Abstract

Generalized bending equations and an example of their use for a radial gradient-index lens system are described. It is shown that the equations are effective tools in real optical design.

© 1990 Optical Society of America

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Figures (3)

Fig. 1

Generalized bending parameters of the system.

Fig. 2

Configuration of the initial zoom lens at the shortest focal length position. Focal length = 38.5–68.5 mm, f/4.5–5.6.

Fig. 3

Configuration of the modified zoom lens at the shortest focal length position.

Tables (2)

Table I First- and Third-Order Properties of the Initial Zoom Lens at the Shortest Focal Length Position

Table II First and Third-Order Properties of the Modified Zoom Lens at the Shortest Focal Length Position

Equations (17)

$y 3 ¯ = y 3 , u 3 ¯ = u 3 for any y 1 and u 1 ,$
$S ( Z i ) = { sinh ( b i Z i ) / b i for n i 1 > 0 , sin ( b i Z i ) / b i for n i 1 < 0 , Z i for n i 1 = 0 ,$
$C ( Z i ) = { cosh ( b i Z i ) for n i 1 > 0 , cos ( b i Z i ) for n i 1 < 0 , 1 for n i 1 = 0 ,$
$Z i = t i i + 1 / n i 0 and b i = { 2 n i 0 n i 1 for n i 1 > 0 , - 2 n i 0 n i 1 for n i 1 < 0 ( i = 1 , 2 ) ,$
$n 10 u 1 ′ = n 10 u 1 - ϕ 1 y 1 , y 2 = y 1 C ( Z 1 ) + n 10 u 1 ′ S ( Z 1 ) , n 10 u 2 = y 1 sgn ( n 11 ) b 1 2 S ( Z 1 ) + n 10 u 1 ′ C ( Z 1 ) , n 20 u 2 ′ = n 10 u 2 - ϕ 2 y 2 , y 3 = y 2 C ( Z 2 ) + n 20 u 2 ′ S ( Z 2 ) , n 20 u 3 = y 2 sgn ( n 21 ) b 2 2 S ( Z 2 ) + n 20 u 2 ′ C ( Z 2 ) . }$
$sng ( x ) = { 1 ( for x > 0 ) , 0 ( for x ¯ = 0 ) , - 1 ( for x ¯ < 0 ) .$
$y 3 = ( [ C ( Z 1 ) - ϕ 1 S ( Z 1 ) ] C ( Z 2 ) - { ( ϕ 1 + ϕ 2 ) C ( Z 1 ) - [ sgn ( n 11 ) b 1 2 + ϕ 1 ϕ 2 ] S ( Z 1 ) } S ( Z 2 ) ) y 1 + { S ( Z 1 ) C ( Z 2 ) + [ C ( Z 1 ) - ϕ 2 S ( Z 1 ) ] S ( Z 2 ) } n 00 u 1 ,$
$n 20 u 3 = - ( { ( ϕ 1 + ϕ 2 ) C ( Z 1 ) - [ sgn ( n 11 ) b 1 2 + ϕ 1 ϕ 2 ] S ( Z 1 ) } C ( Z 2 ) - sgn ( n 21 ) b 2 2 [ C ( Z 1 ) - ϕ 1 S ( Z 1 ) ] S ( Z 2 ) ) y 1 + { [ C ( Z 1 ) - ϕ 2 S ( Z 1 ) ] C ( Z 2 ) + sgn ( n 21 ) b 2 2 S ( Z 1 ) S ( Z 2 ) } n 00 u 1 .$
$y 3 ¯ = ( [ C ( Z 1 ¯ ) - ϕ 1 ¯ S ( Z 1 ¯ ) ] C ( Z 2 ¯ ) - { ( ϕ 1 ¯ + ϕ 2 ¯ ) C ( Z 1 ¯ ) - [ sgn ( n 11 ¯ ) b 1 2 ¯ + ϕ 1 ϕ 2 ¯ ] S ( Z 1 ¯ ) } S ( Z 2 ¯ ) ) y 1 + { S ( Z 1 ¯ ) C ( Z 2 ¯ ) + [ C ( Z 1 ¯ ) - ϕ 2 ¯ S ( Z 1 ¯ ) ] S ( Z 2 ¯ ) } n 00 u 1 ,$
$n 20 u 3 ¯ = - ( { ( ϕ 1 ¯ + ϕ 2 ¯ ) C ( Z 1 ¯ ) - [ sgn ( n 11 ¯ ) b 1 2 ¯ + ϕ 1 ϕ 2 ¯ ] S ( Z 1 ¯ ) } C ( Z 2 ¯ ) - sgn ( n 21 ) b 2 2 [ C ( Z ¯ 1 ) - ϕ ¯ 1 S ( Z 1 ¯ ) ] S ( Z 2 ¯ ) ) y 1 + { [ C ( Z 1 ¯ ) - ϕ 2 ¯ S ( Z 1 ¯ ) ] C ( Z 2 ¯ ) + sgn ( n 21 ) b 2 2 S ( Z 1 ¯ ) S ( Z 2 ¯ ) } n 00 u 1 .$
$[ ( C ( Z 1 ) - ϕ 1 S ( Z 1 ) ] C ( Z 2 ) - { ( ϕ 1 + ϕ 2 ) C ( Z 1 ) - [ sgn ( n 11 ) b 1 2 + ϕ 1 ϕ 2 ] S ( Z 1 ) } S ( Z 2 ) ] y 1 + [ S ( Z 1 ) C ( Z 2 ) + [ C ( Z 1 ) - ϕ 2 S ( Z 1 ) ] S ( Z 2 ) ] n 00 u 1 = ( [ C ( Z 1 ¯ ) - ϕ 1 ¯ S ( Z 1 ¯ ) ] C ( Z 2 ¯ ) - { ( ϕ 1 ¯ + ϕ 2 ¯ ) C ( Z 1 ¯ ) - [ sgn ( n 11 ¯ ) b 1 2 ¯ + ϕ 1 ϕ 2 ¯ ] S ( Z 1 ¯ ) } S ( Z 2 ¯ ) ) y 1 + { S ( Z 1 ¯ ) C ( Z 2 ¯ ) + [ C ( Z 1 ¯ ) - ϕ 2 ¯ S ( Z 1 ¯ ) ] S ( Z 2 ¯ ) } n 00 u 1 .$
$- ( { ( ϕ 1 + ϕ 2 ) C ( Z 1 ) - [ sgn ( n 11 ) b 1 2 + ϕ 1 ϕ 2 ] S ( Z 1 ) } C ( Z 2 ) - sgn ( n 21 ) b 2 2 [ C ( Z 1 ) - ϕ 1 S ( Z 1 ) ] S ( Z 2 ) ) y 1 + { [ C ( Z 1 ) - ϕ 2 S ( Z 1 ) ] C ( Z 2 ) + sgn ( n 21 ) b 2 2 S ( Z 1 ) S ( Z 2 ) } n 00 u 1 = - ( { ( ϕ 1 ¯ + ϕ 2 ¯ ) C ( Z 1 ¯ ) - [ sgn ( n 11 ¯ ) b 1 2 ¯ + ϕ 1 ϕ 2 ¯ ] S ( Z 1 ¯ ) } C ( Z 2 ¯ ) - sgn ( n 21 ) b 2 2 [ C ( Z 1 ¯ ) - ϕ 1 ¯ S ( Z 1 ¯ ) ] S ( Z 2 ¯ ) ) y 1 + { [ C ( Z 1 ¯ ) - ϕ 2 ¯ S ( Z 1 ¯ ) ] C ( Z 2 ¯ ) + sgn ( n 21 ) b 2 2 S ( Z 1 ¯ ) S ( Z 2 ¯ ) } n 00 u 1 .$
$S ( Z 1 ¯ ) C ( Z 2 ¯ ) + [ C ( Z 1 ¯ ) - ϕ 2 ¯ S ( Z 1 ¯ ) ] S ( Z 2 ¯ ) = S ( Z 1 ) C ( Z 2 ) + [ C ( Z 1 ) - ϕ 2 S ( Z 1 ) ] S ( Z 2 ) ,$
$[ C ( Z 1 ¯ ) - ϕ 2 ¯ S ( Z 1 ¯ ) ] C ( Z 2 ¯ ) + sgn ( n 21 ) b 2 2 S ( Z 1 ¯ ) S ( Z 2 ¯ ) = [ C ( Z 1 ) - ϕ 2 S ( Z 1 ) ] C ( Z 2 ) + sgn ( n 21 ) b 2 2 S ( Z 1 ) S ( Z 2 ) .$
$S ( Z 1 ¯ ) = S ( Z 1 ) C ( Z 2 - Z 2 ¯ ) + [ C ( Z 1 ) - ϕ 2 S ( Z 1 ) ] S ( Z 2 - Z 2 ¯ ) .$
$Q 1 ϕ 1 ¯ = Q 1 ϕ 1 + C ( Z 1 ¯ ) C ( Z 2 ¯ ) - C ( Z 1 ) C ( Z 2 ) - [ ϕ 2 C ( Z 1 ¯ ) - sgn ( n 11 ¯ ) b 1 2 S ( Z 1 ¯ ) ] S ( Z 2 ¯ ) + [ ϕ 2 C ( Z 1 ) - sgn ( n 11 ) b 1 2 S ( Z 1 ) ] S ( Z 2 ) ,$
$ϕ 2 ¯ = ( { S ( Z 1 ¯ ) C ( Z 2 ¯ ) - S ( Z 1 ) C ( Z 2 ) - [ C ( Z 1 ) - ϕ 2 S ( Z 1 ) ] S ( Z 2 ) } / S ( Z 2 ¯ ) + C ( Z 1 ¯ ) ) / S ( Z 1 ¯ ) .$