Abstract

The general transformation properties of aberrations in cylindrically symmetric optical systems are described using matrix analysis. The aberrations in an optical system can change with any change in the position of the aperture stop, or when the system operates at new conjugates even if the relative positions of the refractive surfaces and their powers are fixed. Expanding the wavefront aberration function in terms of aberration coefficients allows the new aberration coefficients to be written as linear combinations of the old aberration coefficients for every order. A pattern is established by which higher-order aberration transformations can be calculated.

© 2011 Optical Society of America

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References

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  1. S. Bará, J. Arines, J. Ares, and P. Prado, “Direct transformation of Zernike eye aberration coefficients between scaled, rotated, and/or displaced pupils,” J. Opt. Soc. Am. A 23, 2061–2066(2006).
    [CrossRef]
  2. H. A. Buchdahl, An Introduction to Hamiltonian Optics(Dover, 1963), pp. 36–38.
  3. E. Delano, “First-order design and the y, y¯ diagram,” Appl. Opt. 2, 1251–1256 (1963).
    [CrossRef]
  4. D. Malacara and Z. Malacara, Handbook of Optical Design, 2nd ed. (Marcel Dekker, 2004), p. 76.
  5. J. M. Geary, Introduction to Lens Design with Practical ZEMAX Examples (Willman-Bell, 2007), p. 58.
  6. D. Malacara and Z. Malacara, Handbook of Optical Design, 2nd ed. (Marcel Dekker, 2004), p. 67.

2006 (1)

1963 (1)

Ares, J.

Arines, J.

Bará, S.

Buchdahl, H. A.

H. A. Buchdahl, An Introduction to Hamiltonian Optics(Dover, 1963), pp. 36–38.

Delano, E.

Geary, J. M.

J. M. Geary, Introduction to Lens Design with Practical ZEMAX Examples (Willman-Bell, 2007), p. 58.

Malacara, D.

D. Malacara and Z. Malacara, Handbook of Optical Design, 2nd ed. (Marcel Dekker, 2004), p. 67.

D. Malacara and Z. Malacara, Handbook of Optical Design, 2nd ed. (Marcel Dekker, 2004), p. 76.

Malacara, Z.

D. Malacara and Z. Malacara, Handbook of Optical Design, 2nd ed. (Marcel Dekker, 2004), p. 76.

D. Malacara and Z. Malacara, Handbook of Optical Design, 2nd ed. (Marcel Dekker, 2004), p. 67.

Prado, P.

Appl. Opt. (1)

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

Other (4)

H. A. Buchdahl, An Introduction to Hamiltonian Optics(Dover, 1963), pp. 36–38.

D. Malacara and Z. Malacara, Handbook of Optical Design, 2nd ed. (Marcel Dekker, 2004), p. 76.

J. M. Geary, Introduction to Lens Design with Practical ZEMAX Examples (Willman-Bell, 2007), p. 58.

D. Malacara and Z. Malacara, Handbook of Optical Design, 2nd ed. (Marcel Dekker, 2004), p. 67.

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Tables (3)

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Table 1 ZEMAX Wavefront Aberration Summary for Baseline System before Perturbation

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Table 2 Screen Shot of Seidel Coefficients Calculated from ZEMAX for a + 0.25 mm Stop Shifted Optical System

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Table 3 Aberration Transformation Results Comparison

Equations (71)

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( y ¯ * y * ) ( a b c d ) ( y ¯ y ) ,
( a b c d )
( r ω ) ( y ¯ y υ ¯ υ ) ( h ρ ) ,
r = y ¯ h + y ρ ,
r = ( h ρ ) ( y ¯ y ) .
h · h = h 2 ,
ρ · ρ = ρ 2 ,
h · ρ = h ρ cos ϕ ,
h * · h * = | h * | 2 ,
ρ * · ρ * = | ρ * | 2 ,
h * · ρ * = h * ρ * cos ϕ .
W = L , M , N = 0 W L , M , N = | h · h | L | ρ · ρ | M | h · ρ | N ,
W = l , m , n W l , m , n = h l ρ m cos n ϕ .
[ B n ] = [ T n ] [ A n ] ,
( cos α sin α sin α cos α ) .
( 1 tan β 0 1 ) .
( 1 0 tan δ 1 ) .
( m 0 m 1 / m ) ,
( a b c d ) .
Det ( a b c d ) = 1.
r = ( h ρ ) ( y ¯ y ) = ( h ρ ) ( d b c a ) ( a b c d ) ( y ¯ y ) = ( h * ρ * ) ( y ¯ * y * ) ,
( h * ρ * ) = ( h ρ ) ( d b c a ) ,
( h ρ ) = ( h * ρ * ) ( a b c d ) .
( h ρ ) = ( a c b d ) ( h * ρ * ) ,
( h x h y ρ x ρ y ) = ( a c b d ) ( h x * h y * ρ x * ρ y * ) | h × ρ | .
W 2 = W 020 ρ 2 + W 111 h ρ cos ϕ + W 200 h 2 .
W 2 = ( W 020 W 111 W 200 ) ( ρ 2 h · ρ h 2 ) .
W = W * .
W 2 = W 2 * ,
( W 020 W 111 W 200 ) ( ρ 2 h · ρ h 2 ) = ( W 020 * W 111 * W 200 * ) ( ( ρ * ) 2 h * · ρ * ( h * ) 2 ) .
h = a h * + c ρ * ,
ρ = b h * + d ρ * .
( W 020 * W 111 * W 200 * ) = ( W 020 W 111 W 200 ) ( d 2 2 b d b 2 c d b c + a d a b c 2 2 a c a 2 ) .
( W 020 * W 111 * W 200 * ) = I 2 ( W 020 W 111 W 200 ) ,
I 2 = ( d 2 c d c 2 2 b d b c + a d 2 a c b 2 a b a 2 ) .
W 3 = ( W 040 W 131 W 220 W 222 W 311 W 400 ) ( ρ 4 ρ 2 ( h · ρ ) h 2 ρ 2 ( h · ρ ) 2 h 2 ( h · ρ ) h 4 ) .
( ρ 4 ρ 2 ( h · ρ ) h 2 ρ 2 ( h · ρ ) 2 h 2 ( h · ρ ) h 4 ) = I 3 i ( ρ * 4 ρ * 2 ( h * · ρ * ) h * 2 ρ * 2 ( h * · ρ * ) 2 h * 2 ( h * · ρ * ) h * 4 ) ,
I 3 i = ( d 4 4 b d 3 2 b 2 d 2 4 b 2 d 2 4 b 3 d b 4 c d 3 a d 3 + 3 b c d 2 b 2 c d + a b d 2 2 b 2 c d + 2 a b d 2 3 a b 2 d + b 3 c a b 3 c 2 d 2 2 a c d 2 + 2 b c 2 d b 2 c 2 + a 2 d 2 4 a b c d 2 a b 2 c + 2 a 2 b d a 2 b 2 c 2 d 2 2 a c d 2 + 2 b c 2 d 2 a b c d b 2 c 2 + a 2 d 2 + 2 a b c d 2 a b 2 c + 2 a 2 b d a 2 b 2 c 3 d 3 a c 2 d + b c 3 a b c 2 + a 2 c d 2 a b c 2 + 2 a 2 c d 3 a 2 b c + a 3 d a 3 b c 4 4 a c 3 2 a 2 c 2 4 a 2 c 2 4 a 3 c a 4 ) .
( ρ 4 ρ 2 ( h · ρ ) h 2 · ρ 2 + 2 ( h · ρ ) 2 h 2 ( h · ρ ) ρ 4 ) = I 31 ( ρ * 4 ρ * 2 ( h * · ρ * ) h * 2 · ρ * 2 + 2 ( h * · ρ * ) 2 h * 2 ( h * · ρ * ) ρ * 4 ) ,
( d 4 4 b d 3 2 b 2 d 2 4 b 3 d b 4 c d 3 a d 3 + 3 b c d 2 b 2 c d + a b d 2 3 a b 2 d + b 3 c a b 3 c 2 d 2 2 a c d 2 + 2 b c 2 d b 2 c 2 + a 2 d 2 + 4 a b c d 2 a b 2 c + 2 a b 2 d a 2 b 2 c 3 d 3 a c 2 d + b c 3 a b c 2 + a 2 c d 3 a 2 b c + a 3 d a 3 b c 4 4 a c 3 2 a 2 c 2 4 a 3 c a 4 ) .
W 3 R = ( W 040 W 131 A 1 W 220 + A 2 W 222 W 311 W 400 ) ( ρ 4 ρ 2 ( h · ρ ) h 2 ρ 2 + 2 ( h · ρ ) 2 h 2 ( h · ρ ) h 4 ) ,
W 3 = W 3 R + M X 3 ,
M X 3 = ( A 3 · W 220 + A 4 · W 222 ) ( h 2 ρ 2 ( h · ρ ) 2 ) .
( W 040 W 131 W 220 W 222 W 311 W 400 ) ( ρ 4 ρ 2 ( h · ρ ) h 2 ρ 2 ( h · ρ ) 2 h 2 ( h · ρ ) h 4 ) = ( W 040 W 131 A 1 W 220 + A 2 W 222 W 311 W 400 ) ( ρ 4 ρ 2 ( h · ρ ) h 2 ρ 2 + 2 ( h · ρ ) 2 h 2 ( h · ρ ) h 4 ) + M X 3 .
A 1 = A 2 = 1 3 ,
A 3 = A 4 = 2 3 .
W 3 R = W 3 R * ,
( W 040 * W 131 * W 220 * + W 222 * W 311 * W 400 * ) = I 3 f ( W 040 W 131 W 220 + W 222 W 311 W 400 ) ,
I 3 f = ( d 4 c d 3 c 2 d 2 c 3 d c 4 4 b d 3 a d 3 + 3 b c d 2 2 a c d 2 + 2 b c 2 d 3 a c 2 d + b c 3 4 a c 3 6 b 2 d 2 3 b 2 c d + 3 a b d 2 b 2 c 2 + a 2 d 2 + 4 a b c d 3 a b c 2 + 3 a 2 c d 6 a 2 c 2 4 b 3 d 3 a 2 b d + b 3 c 2 a b 2 c + 2 a 2 b d 3 a 2 b c + a 3 d 4 a c 3 b 4 a b 3 a 2 b 2 a 3 b a 4 ) .
M X 4 = M X 4 C · ( h 2 ρ 2 ( h · ρ ) 2 ) · ( ρ 2 h · ρ h 2 ) ,
( ρ 2 h · ρ h 2 )
M X 4 C = ( B 7 W 240 + B 8 W 242 B 9 W 333 + B 10 W 331 B 11 W 420 + B 12 W 422 ) .
( W 060 * W 151 * W 240 * + W 242 * W 333 * + W 331 * W 420 * + W 422 * W 511 * W 600 * ) = I 4 f ( W 060 W 151 W 240 + W 242 W 333 + W 331 W 420 + W 422 W 511 W 600 ) .
( 1 m 2 0 0 0 1 0 0 0 m 2 ) .
( 1 m 4 0 0 0 0 0 1 m 2 0 0 0 0 0 1 0 0 0 0 0 m 2 0 0 0 0 0 m 4 ) .
( 1 m 6 0 0 0 0 0 0 0 1 m 4 0 0 0 0 0 0 0 1 m 2 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 m 2 0 0 0 0 0 0 0 m 4 0 0 0 0 0 0 0 m 6 ) .
( 1 0 0 2 tan β 1 0 tan 2 β tan β 1 ) .
( 1 0 0 0 0 4 tan β 1 0 0 0 6 tan 2 β 3 tan β 1 0 0 4 tan 3 β 3 tan 2 β 2 tan β 1 0 tan 4 β tan 3 β tan 2 β tan β 1 ) .
( 1 0 0 0 0 0 0 6 tan β 1 0 0 0 0 0 3 tan 2 β 2 tan β 1 0 0 0 0 4 tan 3 β 2 tan 2 β 4 tan β 1 0 0 0 3 tan 4 β 2 tan 3 β 6 tan 2 β 3 tan β 1 0 0 6 tan 5 β 5 tan 4 β 4 tan 3 β 3 tan 2 β 2 tan β 1 0 tan 6 β tan 5 β tan 4 β tan 3 β tan 2 β tan β 1 ) .
k = R n 1 0 tan k β ,
k = R n 2 0 ( k + 1 ) tan k β .
k = R n p 1 0 tan k β ,
W 6 = M 6 C · M 6 V ,
M 6 C = ( W 0100 W 191 W 280 W 820 W 911 W 1000 ) .
Q ¯ = y ¯ * y ¯ y ,
β = tan 1 Q ¯ .
W = ( W 020 W 111 ) = ( 0.6947 0.0400 ) ,
( a b c d ) = ( 1 Q ¯ 0 1 ) = ( 1 0.0304 0 1 ) .
( W 020 * W 111 * ) = ( 1 0 0.0608 1 ) ( 0.6947 0.0400 ) = ( 0.6947 0.0023 ) .
( W 040 * W 131 * W 220 * + W 222 * W 311 * ) = ( 1 0 0 0 0.1217 1 0 0 0.0056 0.0913 1 0 0.0001 0.0028 0.0608 1 ) × ( W 040 * W 131 * W 220 * + W 222 * W 311 * ) ,
( W 040 * W 131 * W 220 * + W 222 * W 311 * ) = ( 1.5819 1.9070 4.0239 1.9028 ) .

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