Abstract

It is well-known that the change in the distance from the optical system to the object affects the image quality. Optical measurement systems, which are aberration-free for a specified position of the measured object, are then limited by induced aberrations for other object positions due to the dependence of aberrations on the varying object position. The consequence of this effect is a change in measurement accuracy. Our work provides a theoretical analysis of the influence of aberrations, which are induced by the change in the object position, on the accuracy of optical measuring systems. Equations were derived for determination of the relative measurement error for monochromatic and polychromatic light using the dependence of the third-order aberrations on the object position. Both geometrical and diffraction theory is used for the analysis. The described effect is not removable in principle and it is necessary to take account to it in high accuracy measurements. Errors can be reduced by a proper design of optical measuring systems. The proposed analysis can be used for measurement corrections.

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References

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2008 (1)

2007 (1)

A. J. E. M. Janssen, S. van Haver, J. J. M. Braat, and P. Dirksen, “Strehl ratio and optimum focus for high-numerical-aperture beams,” J. Eur. Opt. Soc. Rapid Publ. 2, 07008 (2007).
[CrossRef]

1995 (1)

1994 (1)

1989 (1)

1983 (1)

1982 (1)

1968 (1)

1965 (1)

1952 (1)

C. G. Wynne, “Primary aberrations and conjugate change,” Proc. Phys. Soc. 65B, 429–437 (1952).

1939 (1)

Braat, J. J. M.

A. J. E. M. Janssen, S. van Haver, J. J. M. Braat, and P. Dirksen, “Strehl ratio and optimum focus for high-numerical-aperture beams,” J. Eur. Opt. Soc. Rapid Publ. 2, 07008 (2007).
[CrossRef]

Develis, J. B.

Dirksen, P.

A. J. E. M. Janssen, S. van Haver, J. J. M. Braat, and P. Dirksen, “Strehl ratio and optimum focus for high-numerical-aperture beams,” J. Eur. Opt. Soc. Rapid Publ. 2, 07008 (2007).
[CrossRef]

Herzberger, M.

Janssen, A. J. E. M.

A. J. E. M. Janssen, S. van Haver, J. J. M. Braat, and P. Dirksen, “Strehl ratio and optimum focus for high-numerical-aperture beams,” J. Eur. Opt. Soc. Rapid Publ. 2, 07008 (2007).
[CrossRef]

King, W. B.

Mahajan, V. N.

Mikš, A.

Novák, J.

Novák, P.

van Haver, S.

A. J. E. M. Janssen, S. van Haver, J. J. M. Braat, and P. Dirksen, “Strehl ratio and optimum focus for high-numerical-aperture beams,” J. Eur. Opt. Soc. Rapid Publ. 2, 07008 (2007).
[CrossRef]

Walther, A.

Wynne, C. G.

C. G. Wynne, “Primary aberrations and conjugate change,” Proc. Phys. Soc. 65B, 429–437 (1952).

Appl. Opt. (3)

J. Eur. Opt. Soc. Rapid Publ. (1)

A. J. E. M. Janssen, S. van Haver, J. J. M. Braat, and P. Dirksen, “Strehl ratio and optimum focus for high-numerical-aperture beams,” J. Eur. Opt. Soc. Rapid Publ. 2, 07008 (2007).
[CrossRef]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (1)

Proc. Phys. Soc. (1)

C. G. Wynne, “Primary aberrations and conjugate change,” Proc. Phys. Soc. 65B, 429–437 (1952).

Other (9)

http://www.schneiderkreuznach.com/pdf/div/optical_measurement_techniques_with_telecentric_lenses.pdf

E. L. O'Neill, Introduction to Statistical Optics (Addison-Wesley Inc., 1963).

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point-spread function ” in Progress in Optics, E. Wolf, ed. (Elsevier, 2008), Vol. 51, pp. 349–468.

N. Suga, Metrology Handbook: The Science of Measurement (Mitutoyo Ltd., 2007).

T. Yoshizawa, Handbook of Optical Metrology: Principles and Applications (CRC Press, 2009).

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

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge University Press, 1970).

W. T. Welford, Aberrations of the Symmetrical Optical Systems (Academic Press, 1974).

M. Herzberger, Modern Geometrical Optics (Interscience, 1958).

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Figures (2)

Fig. 1
Fig. 1

Imaging of two different planes by optical system.

Fig. 2
Fig. 2

Spot diagrams for different values of transverse magnification m.

Tables (4)

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Table 1 Achromatic Objective Lens (f' = 100 mm, F = 10)

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Table 2 Aberrations – Exact and Approximate Calculation Example 4

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Table 3 Parameters of the Optical System

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Table 4 Comparison of Results Between Derived Equations and Exact Calculations with ZEMAX Software

Equations (66)

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δ y = m δ y + k ( a 1 g 4 S I a 2 g 3 g p S I I + a 3 g 2 g P 2 S I I I + a 4 S I V a 5 g g P 3 S V ) , δ x = m δ x + k ( b 1 g 4 S I b 2 g 3 g p S I I + b 3 g 2 g P 2 S I I I + b 4 S I V ) ,
k = 1 2 n g p 1 3 , a 1 = y P 1 ( y P 1 2 + x P 1 2 ) ,   a 2 = ( 3 y P 1 2 + x P 1 2 ) y ,   a 3 = 3 y P 1 y 2 ,   a 4 = n 2 y P 1 y 2 p 1 2 ,   a 5 = y 3 , b 1 = x P 1 ( y P 1 2 + x P 1 2 ) ,   b 2 = 2 y P 1 x P 1 y ,   b 3 = x P 1 y 2 ,   b 4 = n 2 x P 1 y 2 p 1 2 ,
S = A S 0 ,
S = ( g 4 S I g P g 3 S I I g P 2 g 2 S I I I S I V g P 3 g S V ) , A = ( a 11 a 12 a 13 a 14 a 15 a 16 a 17 0 1 4 a 12 1 2 a 13 1 2 a 14 3 4 a 15 a 16 a 27 0 0 1 6 a 13 0 1 2 a 15 a 16 a 37 0 0 0 1 0 0 0 0 0 0 0 1 4 a 15 a 16 0 ) , S 0 = ( S I S I I S I I I S I V S V S V I 1 ) .
a 11 = ( g P g ) 4 , a 12 = 4 ( g P g ) 3 g P , a 13 = 6 ( g P g ) 2 g P 2 , a 14 = 2 n 2 f 2 ( g P g ) 2 ,
a 15 = 4 ( g P g ) g P 3 , a 16 = g P 4 , a 17 = n f ( g P g ) [ 3 ( g P 2 1 ) 3 g P ( g P g ) + ( g P g ) 2 ] ,
a 27 = n f ( g P g ) [ 2 ( g P 2 1 ) g P ( g P g ) ] , a 37 = n f ( g P g ) ( g P 2 1 ) ,
S = B G ,
B = ( b 11 b 12 b 13 b 14 b 15 0 b 22 b 23 b 24 b 25 0 0 b 33 b 34 b 35 0 0 0 0 b 45 0 0 0 b 54 b 55 ) , G = ( g 4 g 3 g 2 g 1 ) ,
b 11 = S I , b 12 = 4 g P ( S I + S I I ) n f , b 13 = 6 g P 2 ( S I 2 S I I + S I I I ) + 2 n 2 f 2 S I V ,
b 14 = 4 g P 3 ( S I + 3 S I I 3 S I I I + S V ) 4 n 2 f 2 g P S I V + 3 n f ,
b 15 = g P 4 ( S I 4 S I I + 6 S I I I 4 S V + S V I ) + 2 n 2 f 2 g P 2 S I V + g P n f ( g P 2 3 ) ,
b 22 = g P S I I , b 23 = 3 g P 2 ( S I I + S I I I ) + n f ( n f S I V g P ) ,
b 24 = 3 g P 3 ( S I I 2 S I I I + S V ) 2 n f ( n f g P S I V 1 ) ,
b 25 = g P 4 ( S I I + 3 S I I I 3 S V + S V I ) + n 2 f 2 g P 2 S I V + g P n f ( g P 2 2 ) ,
b 33 = g P 2 S I I I , b 34 = 2 g P 3 ( S V S I I I ) n f ( g P 2 1 ) , b 35 = g P 4 ( S I I I + S V I 2 S V ) + n f g P ( g P 2 1 ) ,
b 45 = S I V , b 54 = g P 3 S V , b 55 = g P 4 ( S V + S V I ) .
S 2 = S 1 B ( G 1 G 2 ).
h 1 = s 1 / g , σ 1 = 1 / g , h P 1 = s P 1 / g P , σ P 1 = 1 / g P ,
h 1 = f , σ 1 = 0,   h P 1 = s P 1 / g P , σ P 1 = 1 / g P ,
S I = S I I = S I I I = S I V = S V = S V I = 0.
δ x = ( 1 / 2 g ) [ ( A Y 2 A X + A X 3 ) S I 0 2 tan w A X A Y S I I 0 + tan 2 w A X S I I I 0 ] , δ y = ( 1 / 2 g ) [ ( A X 2 A Y + A Y 3 ) S I 0 tan w ( A X 2 + 3 A Y 2 ) S I I 0 + 3 tan 2 w A Y S I I I 0 ] ,
S I o = f ( g P g )     [ ( g P + g ) 2 g P g 3 ] , S I I o = f ( g P g ) [ g P ( g P + g ) 2 ] , S I I I o = f ( g P g ) ( g P 2 1 ) .
δ x = 1 π R 2 0 2 π 0 R δ x r   d r   d ϕ = 1 π A M 2 0 2 π 0 A M δ x A   d A   d ϕ = 0 ,
δ y = 1 π R 2 0 2 π 0 R δ y r   d r   d ϕ = 1 π A M 2 0 2 π 0 A M δ y A   d A   d ϕ = 1 2 g A M 2 tan w S I I 0 ,
A x = A sin ϕ , A y = A cos ϕ ,
A = A x 2 + A y 2 = x P 1 2 + y P 1 2 / p 1 = r / p 1 .
A M = 1 2 F 0 ( g g ) P
tan w = y p 1 = y s P s 1 = y f ( g P g ) ,
δ y y = ϕ ( m , m P ) F 0 2 ,
ϕ ( m , m P ) = m m P + m 2 ( 1 2 m P 2 ) 8 ( m P m ) 2 ,
ϕ ( m , ) = m 2 / 4.
S I ( λ ) = S I I ( λ ) = S I I I ( λ ) = S I V ( λ ) = S V ( λ ) = S V I ( λ ) = 0.
d ϕ λ = α ( m P , m ) ( d y λ y ) β ( m P , m ) ( d y P λ y P )
α ( m P , m ) = m P m [ m P m ( 4 m P 2 3 ) ] 8 ( m P m ) 3 , β ( m P , m ) = m P m [ m P m ( 4 m m P 3 ) ] 8 ( m P m ) 3 ,
δ s λ = h 1 2 σ 2 C I ,
d y λ y = 1 f ( m m P ) ( m 2 s 1 2 C I m P m s P s 1 C I I ) = d s λ f ( m P m ) h P 1 h 1 H C I I ,
s 1 2 C I = f 2 ( g P g ) 2 C I s P f ( g P g ) ( C I I + C I I P ) + s P 2 C I P ,
s 1 C I I = f ( g P g ) C I I + s P C I P ,
s 1 2 C I = s P f ( g P g ) C I I P + s P 2 C I P , s 1 C I I = s P C I P .
δ s λ = s P g 2 [ f ( g P g ) C I I P s P C I P ] ,
d y λ y = δ s λ + s P 2 m P m C I P f ( m P m ) .
d y P λ y P = 1 f ( m P m ) ( m P 2 s P 2 C I P m P m s P s 1 C I I P ) ,
C I P = C I P , C I I P = C I I P + s P s 1 ( C I P C I I P ) .
d y P λ y P = m P s P 2 f C I P s P C I I P .
d ϕ λ = α ( m P , m ) ( δ s λ + s P 2 m P m C I P f ( m P m ) ) β ( m P , m ) ( m P s P 2 f C I P s P C I I P ) .
δ x = 2 F W X = m W A X , δ y = 2 F W Y = m W A Y ,
W = 1 m ( δ x d A X + δ y d A Y ) ,
I = 1 k 0 2 ( W 2 ¯ W ¯ 2 ) = 1 k 0 2 E 0 ,
W ¯ = 1 S S W d S ,   and   W 2 ¯ = 1 S S W 2 d S ,
W = W 11 r cos  φ   + W 2 0 r 2 + W 4 0 r 4 + W 31 r 3 cos  φ   + W 22 r 2 cos 2 φ ,
W 20 = ( δ s t + δ s s ) / 2 s 0 8 n F 2 ,   W 11 = δ y z y 0 2 F ,   W 40 = δ s K 16 n F 2 , W 31 = δ y K t 6 F ,   W 22 = δ s t δ s s 16 n F 2 ,
E 0 = W 2 ¯ W ¯ 2 = 1 12 W 20 2 + 1 6 W 20 W 40 + 4 45 W 40 2 + 1 4 W 11 2 + 1 3 W 11 W 31 + 1 8 W 31 2 + 1 6 W 22 2 .
E 0 W 11 = 0 , E 0 W 20 = 0.
W 11 = 2 3 W 31 , W 20 = W 40 .
s 0 = δ s t + δ s s 2 + δ s K 2 , y 0 = δ y Z + 2 9 δ y K t .
W 11 = y 0 2 F = 2 3 W 31 = 2 9 F δ y K t ,
δ y K t = 3 2 g tan w A M 2 S I I 0 .
y 0 y = ψ ( m , m P ) F 0 2 ,
ψ ( m , m P ) = m m P + m 2 ( 1 2 m P 2 ) 12 ( m P m ) 2 .
ϕ ( m , m P ) ψ ( m , m P ) = 3 2 .
s = f 1 2 λ ( f F ) 2 .
ε = | ϕ ( m , 1 ) F 0 2 | = | m 8 F 0 2 ( 1 m ) | | m 8 F 0 2 | = | f 8 F 0 2 ( s ε + f ) | | f 8 F 0 2 s ε | ,
s ε = f 8 F 0 2 ε ,
ε g ( m , m P ) = 100 δ y / y = 100 ϕ ( m , m P ) / F 0 2 .
ε d ( m , m P ) = 100 y 0 / y = 100 ψ ( m , m P ) / F 0 2 .

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