Abstract

In designing lenses with automatic computing machinery it is necessary to have certain quantitative measures of lens performance which may be called performance numbers. In a perfect system all the performance numbers are zero. For practical systems it is the object of the designer to reduce the performance numbers to satisfactorily small quantities. It is the principal purpose of this paper to define a characteristic function suitable for use as a performance number, and to derive equations for computing this function and its derivatives with respect to the parameters of the system. The characteristic function H is defined to be the optical path difference between a principal ray and an arbitrary ray, measured on a sphere of infinite radius centered at the intersection of the principal ray and the focal plane.

In addition performance numbers are developed for the chromatic aberration, distortion, and boundary conditions. A merit function is defined which comprises the individual performance numbers.

© 1957 Optical Society of America

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References

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  1. A. L. M’Aulay and F. D. Cruickshank, Proc. Phys. Soc. (London) 57, 302–310 (1945). See also other papers by these authors in Vol. 57.
    [Crossref]
  2. W. S. S. Blaschke, Optica Acta 3, 10–23 (1956).
    [Crossref]
  3. D. P. Feder, J. Opt. Soc. Am. 41, 630–635 (1951).
    [Crossref]
  4. J. L. Synge, Geometrical Optics (Cambridge University Press, New York, 1937).

1956 (1)

W. S. S. Blaschke, Optica Acta 3, 10–23 (1956).
[Crossref]

1951 (1)

1945 (1)

A. L. M’Aulay and F. D. Cruickshank, Proc. Phys. Soc. (London) 57, 302–310 (1945). See also other papers by these authors in Vol. 57.
[Crossref]

Blaschke, W. S. S.

W. S. S. Blaschke, Optica Acta 3, 10–23 (1956).
[Crossref]

Cruickshank, F. D.

A. L. M’Aulay and F. D. Cruickshank, Proc. Phys. Soc. (London) 57, 302–310 (1945). See also other papers by these authors in Vol. 57.
[Crossref]

Feder, D. P.

M’Aulay, A. L.

A. L. M’Aulay and F. D. Cruickshank, Proc. Phys. Soc. (London) 57, 302–310 (1945). See also other papers by these authors in Vol. 57.
[Crossref]

Synge, J. L.

J. L. Synge, Geometrical Optics (Cambridge University Press, New York, 1937).

J. Opt. Soc. Am. (1)

Optica Acta (1)

W. S. S. Blaschke, Optica Acta 3, 10–23 (1956).
[Crossref]

Proc. Phys. Soc. (London) (1)

A. L. M’Aulay and F. D. Cruickshank, Proc. Phys. Soc. (London) 57, 302–310 (1945). See also other papers by these authors in Vol. 57.
[Crossref]

Other (1)

J. L. Synge, Geometrical Optics (Cambridge University Press, New York, 1937).

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Figures (9)

Fig. 1
Fig. 1

Significance of quantities used in ray tracing. Q0, Q1, ɛ1 are unit vectors along the incident ray, the refracted ray, and the surface normal. T0 and T1 locate the ray intersections with the surfaces and M1 is a perpendicular vector from the vertex to the ray. The figure is schematic; in general the rays are not in the plane of the paper. This comment also applies to the subsequent figures.

Fig. 2
Fig. 2

A reference sphere centered at

Fig. 3
Fig. 3

The principal ray strikes the original focal plane at O′ and a displaced one at Q′. B′ and C′ are the projections of O′ and Q′ onto the arbitrary ray.

Fig. 4
Fig. 4

The wave front AA from an object at infinity strikes the entrance pupil plane.

Fig. 5
Fig. 5

A change in a system parameter to the left of this diagram has caused the ray PP′ to be altered into the position AA′. The lines AP and AP′ are perpendicular to the altered ray.

Fig. 6
Fig. 6

A ray PB and a varied ray PA, making an angle dU0 with each other, strike the second surface a distance Δs1 apart.

Fig. 7
Fig. 7

The curvature c1 is changed to c1c1. A ray strikes the original curve at A, the new one at B. The new refracted ray strikes the original surface at A′.

Fig. 8
Fig. 8

The separation is changed by Δt0. A ray strikes the original surface at A, the changed one at B. A translation of the surface back to its original position makes the new refracted ray appear to come from A′.

Fig. 9
Fig. 9

The index N0 is changed causing both ν0 and ν1 to change.

Tables (1)

Tables Icon

Table I For meridian rays only, the table gives the starting values of the partial derivatives of s, U, and L. Each entry is the partial derivative of the quantity on the left with respect to the variable at the top of the column.

Equations (148)

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c i - 1 ,             c i ,             c i + 1 .
c - 1 ,             c 0 ,             c 1 .
N - 1 = N 0 N 0 = N 1 .
i = 0 , 1 , 2 , F ,             F + 1.
W = S * + N S B .
P = S P * + N R .
S P * - S * = P - W - N ( R - S B ) .
S P * - S * = P - W - N [ R - ( R 2 - h 2 ) 1 2 ] .
R - ( R 2 - h 2 ) 1 2 [ h 2 R + ( R 2 - h 2 ) 1 2 ] as R 0 .
lim R ( S P * - S * ) = P - W .
H Y = - W Y = N ( y - y P ) H Z = - W Z = N z .
H = P - + N [ ( y - y P ) Y + z Z ] .
{ H = lim ( S P * - S * ) as R }
H ¯ - H = N l ( 1 - X X P - Y Y P ) / X P .
T 0 + e 0 Q 0 = t 0 i + M 1
M 1 + p 1 Q 0 = T 1
T 1 + r 1 ɛ 1 = r 1 i
( ɛ 1 × Q 1 ) = ν 1 ( ɛ 1 × Q 0 ) .
x 0 + e 0 X 0 = t 0 + M 1 x ,
Q 0 · T 0 + e 0 = t 0 X 0 ,
p 0 Q 0 · T 0 x 0 X 0 + y 0 Y 0 + z 0 Z 0
ζ 0 x 0 - t 0 .
e 0 = t 0 X 0 - p 0
M 1 x = ζ 0 + e 0 X 0 .
ɛ 1 = i - c 1 T 1 .
p 1 = Q 0 · T 1
Q 0 · ɛ 1 = X 0 - c 1 ( Q 0 · T 1 ) .
ξ 1 = X 0 - c 1 p 1 .
M 1 2 + p 1 2 = T 1 2
or             1 = 1 - 2 c 1 x 1 + c 1 2 T 1 2 2 x 1 = c 1 T 1 2 . }
M 1 x + p 1 X 0 = x 1 .
c 1 2 ( M 1 2 + p 1 2 ) = 2 c 1 x 1 = 2 c 1 M 1 x + 2 c 1 p 1 X 0 .
c 1 2 M 1 2 + ( X 0 - ξ 1 ) 2 = 2 c 1 M 1 x + 2 ( X 0 - ξ 1 ) X 0 ,
ξ 1 2 = X 0 2 - c 1 ( c 1 M 1 2 - 2 M 1 x ) .
A 1 c 1 M 1 2 - 2 M 1 x .
M 1 2 = T 0 2 + e 0 2 + t 0 2 + 2 e 0 p 0 - 2 t 0 x 0 - 2 e 0 t 0 X 0 = ζ 0 2 - e 0 2 + y 0 2 + z 0 2 ,
ξ 1 2 = X 0 2 - c 1 A 1 .
c 1 p 1 ( X 0 + ξ 1 ) = X 0 2 - ξ 1 2 = c 1 A 1 .
p 1 = A 1 / ( X 0 + ξ 1 ) = ( X 0 - ξ 1 ) / c 1 ,
T 1 = T 0 + ( e 0 + p 1 ) Q 0 - t 0 i .
L 0 e 0 + p 1 ,
x 1 = ζ 0 + L 0 X 0 y 1 = y 0 + L 0 Y 0 z 1 = z 0 + L 0 Z 0 } .
( Q 1 · ɛ 1 ) ɛ 1 - Q 1 = ν 1 [ ( Q 0 · ɛ 1 ) ɛ 1 - Q 0 ]
Q 1 = ν 1 Q 0 + g 1 ɛ 1
g 1 Q 1 · ɛ 1 - ν 1 ( Q 0 · ɛ 1 ) ξ 1 - ν 1 ξ 1 .
1 - ξ 1 2 = ν 1 2 ( 1 - ξ 1 2 ) ,
Q 1 = ν 1 Q 0 + g 1 ( i - c 1 T 1 )
X 1 = ν 1 X 0 - g 1 c 1 x 1 + g 1 Y 1 = ν 1 Y 0 - g 1 c 1 y 1 Z 1 = ν 1 Z 0 - g 1 c 1 z 1 } .
Q 1 · T 1 = ν 1 Q 0 · T 1 + g 1 ( x 1 - c 1 T 1 2 )
p 1 = ν 1 p 1 - g 1 x 1
ξ 1 = X 1 - c 1 p 1
u 0 y 0 2 + z 0 2 , v 0 y 0 Y 0 + z 0 Z 0 p 0 - x 0 X 0 u 1 = y 1 2 + z 1 2 , v 1 y 1 Y 0 + z 1 Z 0 p 1 - x 1 X 0 } .
v 1 = v 0 + L 0 ( Y 0 2 + Z 0 2 ) u 1 = u 0 + 2 L 0 v 0 + L 0 2 ( Y 0 2 + Z 0 2 ) = u 0 + L 0 ( v 0 + v 1 ) } .
X 0 , Y 0 ,             p 0 , u 0 ,             x 0 , y 0 .
X 1 , Y 1 ,             p 1 , u 1 ,             x 1 , y 1 . e 0 = t 0 X 0 - p 0 ζ 0 = x 0 - t 0 v 0 = p 0 - x 0 X 0 M 1 x = ζ 0 + e 0 X 0 M 1 2 = ( ζ 0 + e 0 ) ( ζ 0 - e 0 ) + u 0 A 1 = c 1 ( M 1 2 ) - 2 M 1 x ξ 1 2 = X 0 2 - c 1 A 1 p 1 = A 1 / ( X 0 + ξ 1 ) L 0 = e 0 + p 1 x 1 = ζ 0 + L 0 X 0 y 1 = y 0 + L 0 Y 0 v 1 = p 1 - x 1 X 0 u 1 = u 0 + L 0 ( v 0 + v 1 ) ξ 1 2 = ( ν 1 ξ 1 + ν 1 ) ( ν 1 ξ 1 - ν 1 ) + 1 g 1 = ξ 1 - ν 1 ξ 1 p 1 = ν 1 p 1 - g 1 x 1 X 1 = ξ 1 + c 1 p 1 Y 1 = ν 1 Y 0 - g 1 c 1 y 1 .
p 0 = y 0 Y 0
u 0 = y 0 2 + z 0 2
= p 0 + K = 0 F N K L K ,
H = P - + N ( v - y P Y ) ,
H = P - + N [ ( y - y P ) Y + z Z ] .
d H = d P - d + N [ ( y - y P ) d Y + z d Z ] + N [ ( d y - d y P ) Y + Z d z ] .
d = d π + N A B = d π + N [ Y d y + Z d z ]
d P = d π P + N [ Y P d y P ] .
d H = d π P - d π + N [ ( y - y P ) d Y + z d Z ] + N ( Y P - Y ) d y P .
d H ¯ = d H + N d l [ 1 - X X P - Y Y P ] / X P .
d y 1 = ( y 1 y 0 ) d y 0 + ( y 1 Y 0 ) d Y 0 , d Y 1 = ( Y 1 y 1 ) d y 1 + ( Y 1 Y 0 ) d Y 0 .
Y - sin U .
c y sin c s .
d Y = - cos U d U = - X d U
d y = ( cos c s ) d s = ( 1 - c 2 y 2 ) 1 2 d s .
d y = x d s .
s 1 s 0 ,             s 1 U 0 ,             U 1 s 1 ,             U 1 U 0 .
- L 0 d U 0 = A B = ( cos I 1 ) d s ,
s 1 / U 0 = - L 0 / ξ 1 .
( cos I 0 ) d s 0 = A B = cos I 1 d s 1 .
s 1 / s 0 = ξ 0 / ξ 1 .
U 1 = U 0 + I 1 - I 1 ,             sin I 1 = ν 1 sin I 1 ,
d U 1 = d U 0 + d I 1 - d I 1 ,             ξ 1 d I 1 = ν 1 ξ 1 d I 1 .
d U 1 = d U 0 + ( g 1 / ξ 1 ) d I 1 .
I 1 = θ 1 - U 0 = c 1 s 1 - U 0
d I 1 = c 1 d s 1 - d U 0 .
d U 1 = ( ν 1 ξ 1 / ξ 1 ) d U 0 + ( c 1 g 1 / ξ 1 ) d s 1 .
A B = sin I 1 Δ s 1 .
d L 0 = ( sin I 1 ) d s 1 - ( sin I 0 ) d s 0 = α 1 d s 1 - α 0 d s 0 .
Q 0 × M 1 = Q 0 × T 1
ɛ 1 × Q 0 = i × Q 0 + c 1 ( Q 0 × T 1 ) .
m 1 = y 1 X 0 - x 1 Y 0 ,
α 1 = Y 0 + c 1 m 1
d s 1 = ( ξ 0 / ξ 1 ) d s 0 - ( L 0 / ξ 1 ) d U 0 d U 1 = ( ν 1 ξ 1 / ξ 1 ) d U 0 + ( c 1 g 1 / ξ 1 ) d s 1 }
m 1 = y 1 X 0 - x 1 Y 0 ,             α 1 = Y 0 + c 1 m 1 α 1 = ν 1 α 1 ,             d L 0 = α 1 d s 1 - α 0 d s 0 } .
d y = d s F + 1 d Y = - X F d U F }
Δ π = N 0 A B - N 1 A T .
A T = A B ( Q 0 · Q 1 ) = A B ( ν 1 + g 1 ξ 1 ) .
Δ π = - A B N 1 g 1 ξ 1 .
d ξ 1 = - c 1 d p 1 - p 1 d c 1 .
2 ξ 1 d ξ 1 = - A 1 d c 1 - c 1 d A 1 = ( - 2 c 1 M 1 2 + 2 M 1 x ) d c 1 .
c 1 ξ 1 d p 1 = ( c 1 M 1 2 - M 1 x - p 1 ξ 1 ) d c 1 = ( c 1 M 1 2 - ζ 0 - e 0 X 0 - p 1 X 0 + c 1 p 1 2 ) d c 1 = x 1 d c 1 .
π / c 1 = - N 1 g 1 x 1 / c 1 .
ξ 1 Δ s 1 = B T = A B sin ( I 1 - I 1 ) = A B ( α 1 ξ 1 - α 1 ξ 1 ) = A B α 1 g 1 .
s 1 / c 1 = α 1 g 1 x 1 / ( c 1 ξ 1 ξ 1 ) .
d U 1 = d U 0 + ( g 1 / ξ 1 ) d I 1 ,             and             sin I 1 = α 1 = Y 0 + c 1 m 1 .
U 1 / c 1 = ( g 1 m 1 ) / ( ξ 1 ξ 1 ) .
L 0 / t 0 = 1 x / ξ 1 .
Δ π = N 0 A B - N 1 D A .
d x 1 = - d t 0 + X 0 d L 0 ,             d y 1 = Y 0 d L 0 ,             d z 1 = Z 0 d L 0 .
d π = [ N 0 - N 1 ( X 0 X 1 + Y 0 Y 1 + Z 0 Z 1 ) ] d L 0 + N 1 X 1 d t 0 = [ N 0 - N 1 ( g 1 ξ 1 + ν 1 ) ] ( 1 x / ξ 1 ) d t 0 + N 1 X 1 d t 0 = ( - N 1 g 1 1 x + N 1 X 1 ) d t 0 = N 0 X 0 d t 0 .
d s 1 = d y 1 / 1 x = ( Y 0 / ξ 1 ) d t 0 .
d ν 0 = - ( ν 0 / N 0 ) d N 0 ,             d ν 1 = ( ν 1 / N 0 ) d N 0 .
( cos I 1 ) d I 1 = ( sin I 1 ) d ν 1 ,             d U 1 = - d I 1 .
d U 1 = - ( α 1 / ξ 1 ) d ν 1
d U 0 = - ( α 0 / ξ 0 ) d ν 0 .
U 0 / N 0 = ( ν 0 α 0 ) / ( N 0 ξ 0 ) , U 1 / N 0 = - ( ν 1 α 1 ) / ( N 0 ξ 1 ) .
π N 0 = L 0 ,             π t 0 = N 0 X 0 ,             π c 1 = - N 1 g 1 x 1 c 1 .
d s 1 = ( ξ 0 / ξ 1 ) d s 0 - ( L 0 / ξ 1 ) d U 0 d U 1 = ( ν 1 ξ 1 / ξ 1 ) d U 0 + ( c 1 g 1 / ξ 1 ) d s 1 }
m 1 = y 1 X 0 - x 1 Y 0 ,             α 1 = Y 0 + c 1 m 1 α 1 = ν 1 α 1 ,             d L 0 = α 1 d s 1 - α 0 d s 0 } .
d H = d π P - d π + N [ ( y - y P ) d Y + z d Z ] + N ( Y P - Y ) d y P + ( N / X P ) ( 1 - X X P - Y Y P ) d l .
d = i = 0 F N i d L i
c 1 = N 0 x 1 c 1 ξ 1 - N 1 x 1 c 1 ξ 1 + i = 1 F N i ( α i + 1 s i + 1 c 1 - α i s i c 1 ) .
c 1 = N 0 x 1 c 1 ξ 1 - N 1 x 1 c 1 ξ 1 - N 1 α 1 α 1 g 1 x 1 ξ 1 ξ 1 c 1 + N F α F + 1 s F + 1 c 1 .
/ c 1 = - N 1 x 1 g 1 / c 1 + N α ( y / c 1 ) .
Y = C y + Y P Z = C z } ,
d Y = y d C + d Y P .
d C / C = ( d Y - d Y P ) / ( Y - Y P ) .
d Y ( Y - Y P ) ( d C / C ) + d Y P d Z Z ( d C / C ) } .
( y - y P ) d Y + z d Z [ v - y P Y - ( y - y P ) Y P ] × ( d C / C ) + ( y - y P ) d Y P .
φ = φ H + φ C + φ E + φ B + φ g + φ f .
φ H = [ H w H s 0 2 ] 2
H = a 0 s 0 2 + a 1 s 0 3 + a 2 s 0 4 + .
H / s 0 n = a 0 s 0 2 - n + a 1 s 0 3 - n + a 2 s 0 4 - n + .
C = D L - D L P
φ C = C 2 w C 2 s 0 2 .
d C = D d L - D d L P
d C = L d D - L P d D
f - y 0 / Y ,             l - y X / Y
d f / f = - d Y / Y , Y d l = ( y / X ) ( d Y / Y ) - X d y
φ f = [ ( f 0 - f ) / w f ] 2 ,
E = F / f - 1 ,
d E = ( 1 + E ) ( d F / F - d f / f ) d F / F = [ d y P + ( Y P / X P ) d l ] / y P } .
φ E = ( E / w E ) 2
N L δ
φ B [ ( δ - N L ) / w B ] 3 for N L < δ 0 for N L δ } .
d φ B = - 3 N ( δ - N L ) 2 w B - 3 d L for N L < δ 0 for N L δ } .
D max = 0.0785 N - 0.1093 ,             D min = 0.0225 N - 0.0262 ,             N = 1.90.
φ g 1 ( N - 1.9 ) 3 w N - 3 for N > 1.9 0 for N 1.9 φ g 2 ( D - D max ) 3 w D - 3 for D > D max 0 for D D max φ g 3 ( D min - D ) 3 w D - 3 for D < D min 0 for D D min .
O ( y P , 0 )
X 0 - 2 = 1 + ( β - α y 0 ) 2 + α 2 z 0 2 Y 0 = ( β - α y 0 ) X 0 Z 0 = - α z 0 X 0 } .
p 0 = y 0 Y 0 + z 0 Z 0 .
where             = N 0 L - 1 + K = 0 F N K L K L - 1 = 2 β y 0 - α ( y 0 2 + z 0 2 ) X 0 - 1 + ( 1 + β 2 ) 1 2 .