Abstract

Three types of merit functions were applied in automatic design of optical systems: mean square value of wave aberrations; variance of wavefront aberrations; and variance of an MTF based function suggested by H. H. Hopkins. They have proved to be very powerful in the final stages of optimization of well corrected systems and lend themselves readily to automatic control of minimization. The method of application and actual design examples are described.

© 1968 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Maréchal, Rev. Opt. 26, 257 (1947).
  2. R. Barakat, A. Houston, J. Opt. Soc. Amer. 55, 1142 (1965).
    [CrossRef]
  3. H. H. Hopkins, Opt. Acta 13, 343 (1966).
    [CrossRef]
  4. J. Meiron, J. Opt. Soc. Amer. 55, 1105 (1965).
    [CrossRef]

1966 (1)

H. H. Hopkins, Opt. Acta 13, 343 (1966).
[CrossRef]

1965 (2)

J. Meiron, J. Opt. Soc. Amer. 55, 1105 (1965).
[CrossRef]

R. Barakat, A. Houston, J. Opt. Soc. Amer. 55, 1142 (1965).
[CrossRef]

1947 (1)

A. Maréchal, Rev. Opt. 26, 257 (1947).

Barakat, R.

R. Barakat, A. Houston, J. Opt. Soc. Amer. 55, 1142 (1965).
[CrossRef]

Hopkins, H. H.

H. H. Hopkins, Opt. Acta 13, 343 (1966).
[CrossRef]

Houston, A.

R. Barakat, A. Houston, J. Opt. Soc. Amer. 55, 1142 (1965).
[CrossRef]

Maréchal, A.

A. Maréchal, Rev. Opt. 26, 257 (1947).

Meiron, J.

J. Meiron, J. Opt. Soc. Amer. 55, 1105 (1965).
[CrossRef]

J. Opt. Soc. Amer. (2)

R. Barakat, A. Houston, J. Opt. Soc. Amer. 55, 1142 (1965).
[CrossRef]

J. Meiron, J. Opt. Soc. Amer. 55, 1105 (1965).
[CrossRef]

Opt. Acta (1)

H. H. Hopkins, Opt. Acta 13, 343 (1966).
[CrossRef]

Rev. Opt. (1)

A. Maréchal, Rev. Opt. 26, 257 (1947).

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 (7)

Fig. 1
Fig. 1

Wavefront W emerging from an optical system. R and R0 are reference spheres.

Fig. 2
Fig. 2

Catadioptric system a used as initial point for systems optimized by means of wavefront-based merit functions. F/1.2, EFL 200 mm, field 3.5°.

Fig. 3
Fig. 3

Aberration curves of initial system a. — tangential - - - sagittal.

Fig. 4
Fig. 4

Aberration curves of system b optimized by the use of the variance E as a merit function.

Fig. 5
Fig. 5

Aberration curves of system c optimized by the use of the MTF-based function K as a merit function.

Fig. 6
Fig. 6

Aberration curves of system d optimized by the use of the mean square wavefront deformation ϕ ¯ 2 as a merit function.

Fig. 7
Fig. 7

Modulation transfer function of initial system a and optimized systems b, c, and d. —z direction, --- y direction.

Tables (1)

Tables Icon

Table I Ray Aberrations Merit Function ψ and Variance E of Wavefront Aberrations at Three Field Angles of Systems

Equations (30)

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

ϕ 2 ¯ = 1 S S ϕ 2 d S ,
ϕ ¯ = 1 S S ϕ d S .
E = ( Δ ϕ ) 2 ¯ = 1 S S ( ϕ - ϕ ¯ ) 2 d S = ϕ 2 ¯ - 2 ϕ ¯ 1 S S ϕ d S + ( ϕ ¯ ) 2 = ϕ 2 ¯ - ( ϕ ¯ ) 2 .
ϕ 2 ¯ - ( ϕ ¯ ) 2 = ( ϕ 0 + d ) 2 ¯ - ( ϕ 0 + d ¯ ) 2 = ϕ 0 2 ¯ + 2 d ϕ ¯ 0 + d 2 - ( ϕ ¯ 0 + d ) 2 = ϕ 0 2 ¯ - ( ϕ ¯ 0 ) 2 .
R 0 = R - ϕ ¯ .
E = ϕ 0 2 ¯ .
D = 1 - ( 2 π / λ ) 2 E .
M ( s , φ ) = 1 - ( 2 π 2 s 2 / λ 2 ) [ V ¯ 2 - ( V ¯ ) 2 ] = 1 - ( 2 π 2 s 2 / λ 2 ) K ( s , φ ) .
V ( u , v , s , φ ) = ( 1 / s ) { ϕ [ u + ( s / 2 ) , v ] - ϕ [ u - ( s / 2 ) , v ] } ,
ϕ i = j = 1 n a i j x j + ϕ i 0 = 0 ,             i = 1 , 2 , , m ,
ϕ 2 ¯ = 1 m i = 1 m ϕ i 2 ,
ϕ i = j = 1 n a i j x j + ϕ i 0 = 0 ,             i = 1 , 2 , , r ,
ϕ i = ϕ i - 1 r k = 1 r ϕ k = ϕ i - ϕ ¯ l , ϕ i 0 = ϕ i 0 - 1 r k = 1 r ϕ k 0 = ϕ i 0 - ϕ ¯ l 0 .
a i j = a i j - 1 r k = 1 r a k j = a i j - a ¯ l j .
E l = 1 r i = 1 r ( ϕ i - 1 r k = 1 r ϕ k ) 2 = 1 r [ i = 1 r ϕ i 2 - 2 1 r i = 1 r ϕ i k = 1 r ϕ k + ( 1 r k = 1 r ϕ k ) 2 ] = 1 r i = 1 r ϕ i 2 - ( 1 r i = 1 r ϕ i ) 2 ,
E = Σ l μ l E l ,
V i = ( 1 / s ) { ϕ i [ u + ( s / 2 ) , v ] - ϕ i [ u - ( s / 2 ) , v ] }
V i = j = 1 n a i j x j + V i 0 = 0 ,             i = 1 , 2 , , r ,
V i = V i - V ¯ l , V ' i 0 = V i 0 - V ¯ l 0 , a i j = a i j - a ¯ l j .
j = 1 n ( a r a j - m a ¯ r a ¯ j ) x j = - a r ϕ 0 + m a ¯ r ϕ ¯ 0 ,             r = 1 , 2 , , n .
ϕ i = j = 1 n a i j x j + ϕ i 0 = 0 ,             i = 1 , 2 , , m ,
E = 1 m i = 1 m ϕ i 2 - [ 1 m i = 1 m ϕ i ] 2 ,
E x r = 2 m i = 1 m ϕ i ϕ i x r - 2 m 2 i = 1 m ϕ i k = 1 m ϕ k x r = 0 ,             r = 1 , 2 , , n ,
i = 1 m a i r ϕ i - 1 m i = 1 m ϕ i k = 1 m a k r = 0 ,             r = 1 , 2 , , n .
a ¯ r = 1 m k = 1 m a k r ,
i = 1 n ( a i r - a ¯ r ) ϕ i = 0 ,             r = 1 , 2 , , n .
i = 1 m ( a i r - a ¯ r ) ϕ i 0 + i = 1 m ( a i r - a ¯ r ) j = 1 n a i j x j = i = 1 m a i r ϕ i 0 - a ¯ r i = 1 m ϕ i 0 + i = 1 m j = 1 n a i r a i j x j - a ¯ r i = 1 m j = 1 n a i j x j = 0 ,             r = 1 , 2 , , n ,
j = 1 n ( i = 1 m a i r a i j - a ¯ r i = 1 m a i j ) x j = i = 1 m a i r ϕ i 0 + a ¯ r i = 1 m ϕ i 0 ,             r = 1 , 2 , , n .
j = 1 n ( i = 1 m a i r a i j - m a ¯ r a ¯ j ) x j = - i = 1 m a i r ϕ i 0 + m a ¯ r ϕ ¯ 0 ,             r = 1 , 2 , , n .
j = 1 n ( a r a j - m a ¯ r a ¯ j ) x j = - a r ϕ 0 + m a ¯ r ϕ ¯ 0 ,             r = 1 , 2 , , n .

Metrics