Abstract

Our work deals with a theoretical analysis of an influence of a deviation from the sine condition of an optical system on the kinematics of aperture rays, which are transformed by this optical system. An exact formula for calculation of the value of the departure from the sine condition that has to be satisfied by an optical system to achieve a constant speed of transformed rays is derived. Approximate relations valid in the scope of the Seidel third-order aberrations theory are also derived and examples of optical systems that satisfy these conditions are presented. Obtained results are important for a design of scanning optical systems used, e.g., for distance measurements and laser systems for surface refinement of materials.

© 2009 Optical Society of America

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References

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2009

2008

2002

1992

K. S. Chang, S. T. Lu, Y. P. Wu, and C. Chou, “Correction algorithms in a laser scanning dimension measurement system,” IEEE Proc. A 139, 57-60 (1992).

M. Shibuya, “Exact sine condition in the presence of spherical aberration,” Appl. Opt. 31, 2206-2210 (1992).
[CrossRef] [PubMed]

1979

1963

Barakat, R.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Burge, J.

Chang, K. S.

K. S. Chang, S. T. Lu, Y. P. Wu, and C. Chou, “Correction algorithms in a laser scanning dimension measurement system,” IEEE Proc. A 139, 57-60 (1992).

Choi, N.

Chou, C.

K. S. Chang, S. T. Lu, Y. P. Wu, and C. Chou, “Correction algorithms in a laser scanning dimension measurement system,” IEEE Proc. A 139, 57-60 (1992).

Comastri, Silvia A.

Czapski, S.

S. Czapski and O. Eppenstein, Grundzüge der theorie der optischen instrumente (Verlag, 1924).

Eppenstein, O.

S. Czapski and O. Eppenstein, Grundzüge der theorie der optischen instrumente (Verlag, 1924).

Hwang, S.

Kim, J.

Kim, M.

Korn, G. A.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientist and Engineers (Dover, 2000).

Korn, T. M.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientist and Engineers (Dover, 2000).

Lee, Y.

Lev, D.

Lu, S. T.

K. S. Chang, S. T. Lu, Y. P. Wu, and C. Chou, “Correction algorithms in a laser scanning dimension measurement system,” IEEE Proc. A 139, 57-60 (1992).

Mikš, A.

A. Mikš, Applied Optics (Czech Technical University, 2009).
[PubMed]

Ratto, J. O.

Shibuya, M.

Simon, J. M.

Welford, W. T.

W. T. Welford, Aberrations of Optical Systems (Hilger, 1986).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Wu, Y. P.

K. S. Chang, S. T. Lu, Y. P. Wu, and C. Chou, “Correction algorithms in a laser scanning dimension measurement system,” IEEE Proc. A 139, 57-60 (1992).

Zhao, C.

Appl. Opt.

IEEE Proc. A

K. S. Chang, S. T. Lu, Y. P. Wu, and C. Chou, “Correction algorithms in a laser scanning dimension measurement system,” IEEE Proc. A 139, 57-60 (1992).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Express

Other

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

A. Mikš, Applied Optics (Czech Technical University, 2009).
[PubMed]

S. Czapski and O. Eppenstein, Grundzüge der theorie der optischen instrumente (Verlag, 1924).

W. T. Welford, Aberrations of Optical Systems (Hilger, 1986).

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientist and Engineers (Dover, 2000).

www.zemax.com.

www.sinopt.com.

http://www.aeroel.it/en/tecnologia/tech_2.htm.

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Figures (2)

Fig. 1
Fig. 1

Transformation of rays by the optical system.

Fig. 2
Fig. 2

Scheme of the scanning optical system for linear dimensions measurement.

Tables (2)

Tables Icon

Table 1 Values of Function g ( u )

Tables Icon

Table 2 Parameters of Optical System: the First Cemented Doublet

Equations (26)

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

m = n sin u n sin u ,
m 0 = lim u 0 n sin u n sin u = n u 0 n u 0 ,
v = d s d s v = R R ( d u d u ) v ,
d u = ( v R v R ) d u = C 1 d u ,
u = C 1 u + C 2 ,
v 0 v 0 = u 0 R u 0 R ,
C 1 = v R v R = v 0 R v 0 R = u 0 u 0 = n n m 0 .
u 0 = C 1 u 0 + C 2 .
u = C 1 u = ( n n m 0 ) u ,
m = n sin u n sin u = n sin u n sin ( C 1 u ) = n sin ( u / C 1 ) n sin u .
δ m = m m 0 = n sin u n sin ( C 1 u ) n u 0 n u 0 = n sin ( u / C 1 ) n sin u n u 0 n u 0 .
δ f = f ( u ) f 0 = f 0 ( u sin u 1 ) ,
d m d u = m 0 cos u sin u m cos u sin u .
d f d u = f 0 1 sin u f cos u sin u .
f ( u ) = f 0 u sin u ,
d u = ( m m 0 ) sin u m 0 cos u m cos u , d u = ( f f 0 ) sin u f 0 f cos u .
δ m m 0 = n sin ( u / C 1 ) m 0 n sin u 1 1 6 u 2 ( 1 n 2 n 2 m 0 2 ) .
δ m m 0 = u 2 2 n u 0 3 u ¯ 0 ( s s ¯ ) S I I = u 2 2 n ( s s ¯ ) m 0 u 0 3 u ¯ 0 S I I ,
S I I = 1 3 ( 1 m 0 2 ) ( s s ¯ ) m 0 u 0 3 u ¯ 0 .
δ f f 0 1 6 u 2 ,
δ f f 0 = u 2 2 n f 0 u 0 3 u ¯ 0 S I I ,
S I I = 1 3 f 0 u 0 3 u ¯ 0 ,
S I I = 1 3 .
S I = 0 , S I I = 1 / 3 ,
v = ω f 0 g ( u ) ,
g ( u ) = ( 1 + 1 2 sin 2 u ) cos u .

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