Table 1
Wavefront Aberrations
Aberration Vector Form Algebraic Form
j
m
n
Zero-order Uniform piston
W
000
W
000
0 0 0 Second-order Gauss Quadratic piston
W
200
(
H
→
·
H
→
)
W
200
H
2
1 0 0 Magnification
W
111
(
H
→
·
ρ
→
)
W
111
H
ρ
cos
(
ϕ
)
0 1 0 Focus
W
020
(
ρ
→
·
ρ
→
)
W
020
ρ
2
0 0 1 Fourth order Seidel Spherical aberration
W
040
(
ρ
→
·
ρ
→
)
2
W
040
ρ
4
0 0 2 Coma
W
131
(
H
→
·
ρ
→
)
(
ρ
→
·
ρ
→
)
W
131
H
ρ
3
cos
(
ϕ
)
0 1 1 Astigmatism
W
222
(
H
→
·
ρ
→
)
2
W
222
H
2
ρ
2
cos
2
(
ϕ
)
0 2 0 Field curvature
W
220
(
H
→
·
H
→
)
(
ρ
→
·
ρ
→
)
W
220
H
2
ρ
2
1 0 1 Distortion
W
311
(
H
→
·
H
→
)
(
H
→
·
ρ
→
)
W
311
H
3
ρ
cos
(
ϕ
)
1 1 0 Quartic piston
W
400
(
H
→
·
H
→
)
2
W
400
H
4
2 0 0 Sixth order Schwarzschild Oblique spherical aberration
W
240
(
H
→
·
H
→
)
(
ρ
→
·
ρ
→
)
2
W
240
H
2
ρ
4
1 0 2 Coma
W
331
(
H
→
·
H
→
)
(
H
→
·
ρ
→
)
(
ρ
→
·
ρ
→
)
W
331
H
3
ρ
3
cos
(
ϕ
)
1 1 1 Astigmatism
W
422
(
H
→
·
H
→
)
(
H
→
·
ρ
→
)
2
W
422
H
4
ρ
2
cos
2
(
ϕ
)
1 2 0 Field curvature
W
420
(
H
→
·
H
→
)
2
(
ρ
→
·
ρ
→
)
W
420
H
4
ρ
2
2 0 1 Distortion
W
511
(
H
→
·
H
→
)
2
(
H
→
·
ρ
→
)
W
511
H
5
ρ
cos
(
ϕ
)
2 1 0 Piston
W
600
(
H
→
·
H
→
)
3
W
600
H
6
3 0 0 Spherical aberration
W
060
(
ρ
→
·
ρ
→
)
3
W
060
ρ
6
0 0 3 Unnamed
W
151
(
H
→
·
ρ
→
)
(
ρ
→
·
ρ
→
)
2
W
151
H
ρ
5
cos
(
ϕ
)
0 1 2 Unnamed
W
242
(
H
→
·
ρ
→
)
2
(
ρ
→
·
ρ
→
)
W
242
H
2
ρ
4
cos
2
(
ϕ
)
0 2 1 Unnamed
W
333
(
H
→
·
ρ
→
)
3
W
333
H
3
ρ
3
cos
3
(
ϕ
)
0 3 0
Table 2
Seidel Aberration Coefficients
Aberration Seidel Sum
W
040
=
1
8
S
I
S
I
=
−
∑
A
2
y
Δ
(
u
n
)
W
131
=
1
2
S
II
S
II
=
−
∑
A
A
¯
y
Δ
(
u
n
)
W
222
=
1
2
S
III
S
III
=
−
∑
A
¯
2
y
Δ
(
u
n
)
W
220
=
1
4
S
IV
S
IV
=
−
∑
Ψ
2
P
−
∑
A
¯
2
y
Δ
(
u
n
)
W
311
=
1
2
S
V
S
V
=
−
∑
A
¯
A
[
Ψ
2
P
+
A
¯
2
y
Δ
(
u
n
)
]
,
S
V
=
−
∑
A
¯
[
A
¯
2
Δ
(
1
n
2
)
y
−
(
Ψ
+
A
¯
y
)
y
¯
P
]
W
400
=
1
8
S
VI
S
VI
=
−
∑
A
¯
2
y
¯
Δ
(
u
¯
n
)
Table 3
Quantities Derived from Paraxial Data used in Computing Aberration Coefficients
Quantity Formula Refraction-invariant marginal ray
A
=
n
i
=
n
u
+
n
y
c
Refraction- invariant chief ray
A
¯
=
n
i
¯
=
n
u
¯
+
n
y
¯
c
Lagrange invariant
Ψ
=
n
u
¯
y
−
n
u
y
¯
=
A
¯
y
−
A
y
¯
Surface curvature
c
=
1
r
Petzval sum term
P
=
c
·
Δ
(
1
n
)
Table 4
Pupil Aberrations
Name Identity between Pupil and Image Aberration Coefficients Pupil spherical aberration
W
¯
040
=
W
400
Pupil coma
W
¯
131
=
W
311
+
1
2
Ψ
·
Δ
{
u
¯
2
}
Pupil astigmatism
W
¯
222
=
W
222
+
1
2
Ψ
·
Δ
{
u
u
¯
}
Pupil sagittal field curvature
W
¯
220
=
W
220
+
1
4
Ψ
·
Δ
{
u
u
¯
}
Pupil distortion
W
¯
311
=
W
131
+
1
2
Ψ
·
Δ
{
u
2
}
Pupil piston
W
¯
400
=
W
040
Table 5
Extrinsic Coefficients for Combination of Systems A and B
With Aperture Vector
ρ
→
at Entrance Pupil With Aperture Vector
ρ
→
at Exit Pupil
W
060
E
−
=
1
Ψ
{
4
W
040
B
W
¯
311
A
}
W
060
E
+
=
−
1
Ψ
{
4
W
040
A
W
¯
311
B
}
W
151
E
−
=
{
1
Ψ
3
W
131
B
W
¯
311
A
+
8
W
040
B
W
¯
220
A
+
8
W
040
B
W
¯
222
A
}
W
151
E
+
=
−
1
Ψ
{
3
W
131
A
W
¯
311
B
+
8
W
040
A
W
¯
220
B
+
8
W
040
A
W
¯
222
B
}
W
242
E
−
=
1
Ψ
{
2
W
222
B
W
¯
311
A
+
4
W
131
B
W
¯
220
A
+
6
W
131
B
W
¯
222
A
+
8
W
040
B
W
¯
131
A
}
W
242
E
+
=
−
1
Ψ
{
2
W
222
A
W
¯
311
B
+
4
W
131
A
W
¯
220
B
+
6
W
131
A
W
¯
222
B
+
8
W
040
A
W
¯
131
B
}
W
333
E
−
=
1
Ψ
{
4
W
131
B
W
¯
131
A
+
4
W
222
B
W
¯
222
A
}
W
333
E
+
=
−
1
Ψ
{
4
W
131
A
W
¯
131
B
+
4
W
222
A
W
¯
222
B
}
W
240
E
−
=
1
Ψ
{
2
W
131
B
W
¯
220
A
+
2
W
220
B
W
¯
311
A
+
4
W
040
B
W
¯
131
A
}
W
240
E
+
=
−
1
Ψ
{
2
W
131
A
W
¯
220
B
+
2
W
220
A
W
¯
311
B
+
4
W
040
A
W
¯
131
B
}
W
331
E
−
=
1
Ψ
{
5
W
131
B
W
¯
131
A
+
4
W
220
B
W
¯
220
A
+
4
W
220
B
W
¯
222
A
+
4
W
222
B
W
¯
220
A
+
W
311
B
W
¯
311
A
+
16
W
040
B
W
¯
040
A
}
W
331
E
+
=
−
1
Ψ
{
5
W
131
A
W
¯
131
B
+
4
W
220
A
W
¯
220
B
+
4
W
220
A
W
¯
222
B
+
4
W
222
A
W
¯
220
B
+
W
311
A
W
¯
311
B
+
16
W
040
A
W
¯
040
B
}
W
422
E
−
=
1
Ψ
{
2
W
311
B
W
¯
222
A
+
4
W
220
B
W
¯
131
A
+
6
W
222
B
W
¯
131
A
+
8
W
131
B
W
¯
040
A
}
W
422
E
+
=
−
1
Ψ
{
2
W
311
A
W
¯
222
B
+
4
W
220
A
W
¯
131
B
+
6
W
222
A
W
¯
131
B
+
8
W
131
A
W
¯
040
B
}
W
420
E
−
=
1
Ψ
{
2
W
220
B
W
¯
131
A
+
2
W
311
B
W
¯
220
A
+
4
W
131
B
W
¯
040
A
}
W
420
E
+
=
−
1
Ψ
{
2
W
220
A
W
¯
131
B
+
2
W
311
A
W
¯
220
B
+
4
W
131
A
W
¯
040
B
}
W
511
E
−
=
1
Ψ
{
3
W
311
B
W
¯
131
A
+
8
W
220
B
W
¯
040
A
+
8
W
222
B
W
¯
040
A
}
W
511
E
+
=
−
1
Ψ
{
3
W
311
A
W
¯
131
B
+
8
W
220
A
W
¯
040
B
+
8
W
222
A
W
¯
040
B
}
W
600
E
−
=
1
Ψ
{
4
W
311
B
W
¯
040
A
}
W
600
E
+
=
−
1
Ψ
{
4
W
311
A
W
¯
040
B
}
Added terms when the center of the reference sphere is located at the intersection of the chief ray with the Gaussian image plane
Δ
W
331
E
−
=
−
W
311
B
Δ
A
{
u
2
}
/
2
Δ
W
331
E
+
=
+
W
311
A
Δ
B
{
u
2
}
/
2
Δ
W
422
E
−
=
−
W
311
B
Δ
A
{
u
u
¯
}
Δ
W
422
E
+
=
+
W
311
A
Δ
B
{
u
u
¯
}
Δ
W
420
E
−
=
−
W
311
B
Δ
A
{
u
u
¯
}
/
2
Δ
W
420
E
+
=
+
W
311
A
Δ
B
{
u
u
¯
}
/
2
Δ
W
511
E
−
=
−
3
W
311
B
Δ
A
{
u
¯
2
}
/
2
Δ
W
511
E
+
=
+
3
W
311
A
Δ
B
{
u
¯
2
}
/
2
Table 6
Wavefront Change at the Exit Pupil on Propagation in Free Space the Distance
Δ
Z
′
=
−
y
¯
·
u
¯
′
−
1
from the Pupil
a
Δ
Z
′
W
−
(
H
→
,
ρ
→
)
=
1
2
y
¯
y
1
Ψ
∇
ρ
W
(
H
→
,
ρ
→
)
·
∇
ρ
W
(
H
→
,
ρ
→
)
Δ
W
060
−
=
1
2
y
¯
y
1
Ψ
{
16
W
040
W
040
}
Δ
W
151
−
=
1
2
y
¯
y
1
Ψ
{
24
W
040
W
131
}
Δ
W
242
−
=
1
2
y
¯
y
1
Ψ
{
16
W
040
W
222
+
8
W
131
W
131
}
Δ
W
240
−
=
1
2
y
¯
y
1
Ψ
{
16
W
040
W
220
+
W
131
W
131
}
Δ
W
333
−
=
1
2
y
¯
y
1
Ψ
{
8
W
131
W
222
}
Δ
W
331
−
=
1
2
y
¯
y
1
Ψ
{
8
W
040
W
311
+
4
W
131
W
222
+
12
W
131
W
220
}
Δ
W
422
−
=
1
2
y
¯
y
1
Ψ
{
4
W
131
W
311
+
4
W
222
W
222
+
8
W
222
W
220
}
Δ
W
420
−
=
1
2
y
¯
y
1
Ψ
{
4
W
220
W
220
+
2
W
311
W
131
}
Δ
W
511
−
=
1
2
y
¯
y
1
Ψ
{
4
W
222
W
311
+
4
W
220
W
311
}
Δ
W
600
−
=
1
2
y
¯
y
1
Ψ
{
W
311
W
311
}
a
The aperture vector is at the initial propagation plane.
Table 7
Change in Wavefront on Placing the Aperture Vector at the Exit Pupil
a
W
+
(
H
→
,
ρ
→
)
−
W
−
(
H
→
,
ρ
→
)
=
−
1
Ψ
∇
ρ
W
(
H
→
,
ρ
→
)
·
∇
H
W
¯
(
H
→
,
ρ
→
)
W
060
+
−
W
060
−
=
−
1
Ψ
{
4
W
040
W
¯
311
}
W
151
+
−
W
151
−
=
−
1
Ψ
{
3
W
131
W
¯
311
+
8
W
040
W
¯
220
+
8
W
040
W
¯
222
}
W
242
+
−
W
242
−
=
−
1
Ψ
{
2
W
222
W
¯
311
+
4
W
131
W
¯
220
+
6
W
131
W
¯
222
+
8
W
040
W
¯
131
}
W
333
+
−
W
333
−
=
−
1
Ψ
{
4
W
131
W
¯
131
+
4
W
222
W
¯
222
}
W
240
+
−
W
240
−
=
−
1
Ψ
{
2
W
131
W
¯
220
+
2
W
220
W
¯
311
+
4
W
040
W
¯
131
}
W
331
+
−
W
331
−
=
−
1
Ψ
{
5
W
131
W
¯
131
+
4
W
220
W
¯
220
+
4
W
220
W
¯
222
+
4
W
222
W
¯
220
+
W
311
W
¯
311
+
16
W
040
W
¯
040
}
W
422
+
−
W
422
−
=
−
1
Ψ
{
2
W
311
W
¯
222
+
4
W
220
W
¯
131
+
6
W
222
W
¯
131
+
8
W
131
W
¯
040
}
W
420
+
−
W
420
−
=
−
1
Ψ
{
2
W
220
W
¯
131
+
2
W
311
W
¯
220
+
4
W
131
W
¯
040
}
W
511
+
−
W
511
−
=
−
1
Ψ
{
3
W
311
W
¯
131
+
8
W
220
W
¯
040
+
8
W
222
W
¯
040
}
W
600
+
−
W
600
−
=
−
1
Ψ
{
4
W
311
W
¯
040
}
a
The reference sphere is centered at the Gaussian image point.
Table 8
Image and Pupil Coefficient Relationships for a Spherical Surface
4
W
040
y
¯
y
=
W
¯
311
−
W
¯
311
0
4
W
¯
040
y
y
¯
=
W
311
−
W
311
0
W
131
y
¯
y
=
W
¯
222
−
W
¯
311
0
y
¯
y
W
¯
131
y
y
¯
=
W
222
−
W
311
0
y
y
¯
W
222
y
¯
y
=
W
¯
131
+
W
311
0
W
¯
222
y
y
¯
=
W
131
+
W
¯
311
0
W
220
y
¯
y
=
1
2
W
¯
131
−
1
2
W
¯
311
0
(
y
¯
y
)
2
W
¯
220
y
y
¯
=
1
2
W
131
−
1
2
W
311
0
(
y
y
¯
)
2
W
311
y
¯
y
=
4
W
¯
040
+
W
311
0
y
¯
y
W
¯
311
y
y
¯
=
4
W
040
+
W
¯
311
0
y
y
¯
W
311
0
=
W
311
y
=
0
=
1
2
1
R
Ψ
A
¯
y
¯
Δ
{
1
n
}
=
1
2
Ψ
α
¯
(
u
¯
′
−
u
¯
)
W
¯
311
0
=
W
¯
311
y
¯
=
0
=
−
1
2
1
R
Ψ
A
y
Δ
{
1
n
}
=
−
1
2
Ψ
α
(
u
′
−
u
)
α
=
y
r
α
¯
=
y
¯
r
Table 9
Quantities Used in Calculation of Intrinsic Aberration Coefficients with the Aperture Vector
ρ
→
at the Exit Pupil
W
220
P
=
−
1
4
Ψ
2
P
W
240
C
C
+
=
+
1
16
A
r
Ψ
2
Δ
{
u
n
2
}
+
1
8
1
r
Ψ
2
Δ
{
u
2
n
}
+
1
4
y
2
r
2
W
220
P
+
y
r
u
′
W
220
P
−
1
4
u
r
Ψ
2
Δ
{
u
n
}
W
420
C
C
+
=
3
16
1
r
3
Ψ
4
Δ
{
1
n
}
1
A
2
(
W
420
C
C
+
=
0
for
A
=
0
)
W
331
C
C
+
=
−
2
W
220
P
·
u
′
u
¯
′
W
151
C
C
+
=
−
4
W
040
·
u
′
u
¯
′
W
242
C
C
+
=
−
2
W
040
·
u
¯
′
2
Table 10
Intrinsic Aberration Coefficients with the Aperture Vector
ρ
→
at the Exit Pupil
a
W
060
I
+
=
W
040
[
1
2
y
2
r
2
−
1
2
A
(
u
′
n
′
+
u
n
)
+
2
y
r
u
′
]
−
8
Ψ
W
040
·
W
040
y
¯
y
W
151
I
+
=
6
A
¯
A
W
060
I
+
+
W
131
u
′
2
+
W
151
C
C
+
W
242
I
+
=
12
(
A
¯
A
)
2
W
060
I
+
+
7
2
W
222
u
′
2
−
3
W
131
u
′
u
¯
′
+
W
242
C
C
+
W
333
I
+
=
8
(
A
¯
A
)
3
W
060
I
+
+
4
(
A
¯
A
)
2
W
151
C
C
+
+
3
A
¯
A
W
222
u
′
2
+
2
A
¯
A
W
242
C
C
+
+
2
W
222
u
′
u
¯
′
W
240
I
+
=
W
240
C
C
+
+
3
(
A
¯
A
)
2
W
060
I
+
−
8
1
Ψ
A
¯
A
W
040
·
W
220
P
+
W
222
u
′
2
−
W
131
u
′
u
¯
′
W
331
I
+
=
4
A
¯
A
W
240
I
+
+
2
A
¯
A
W
242
C
C
+
+
A
¯
A
W
u
220
′
2
+
W
311
u
'
2
+
W
331
C
C
W
422
I
+
=
4
(
A
¯
A
)
2
W
240
I
+
+
2
A
¯
A
W
331
C
C
+
−
2
W
222
u
¯
′
2
−
W
u
¯
220
′
2
+
2
(
A
¯
A
)
2
W
u
220
′
2
+
W
311
u
′
u
¯
′
+
1
2
A
¯
A
W
311
u
'
2
+
2
A
¯
A
W
u
220
′
u
¯
′
W
420
I
+
=
3
(
A
¯
A
)
4
W
060
I
+
+
2
(
A
¯
A
)
2
W
240
C
C
+
−
4
(
A
¯
A
)
2
1
Ψ
W
131
·
W
220
P
−
2
A
¯
A
1
Ψ
W
220
P
·
W
220
P
+
W
420
C
C
+
+
A
¯
A
W
331
C
C
+
+
(
A
¯
A
)
2
W
220
P
u
′
2
+
1
2
A
¯
A
W
311
u
′
2
−
1
2
W
311
u
′
u
¯
′
+
2
A
¯
A
W
220
P
u
′
u
¯
′
+
(
A
¯
A
)
2
W
222
u
′
2
−
1
2
W
222
u
¯
′
2
W
511
I
+
=
6
(
A
¯
A
)
5
W
060
I
+
+
4
(
A
¯
A
)
3
[
W
240
C
C
+
−
2
1
Ψ
W
131
·
W
220
P
]
+
2
A
¯
A
[
W
420
C
C
+
−
2
A
¯
A
1
Ψ
W
220
P
·
W
220
P
]
+
(
A
¯
A
)
3
W
222
u
′
2
+
2
(
A
¯
A
)
2
W
311
u
′
2
−
(
A
¯
A
)
2
W
222
u
′
u
¯
′
−
2
(
A
¯
A
)
2
W
220
P
u
′
u
¯
′
−
A
¯
A
W
222
u
¯
′
2
+
1
2
W
311
u
¯
′
2
W
600
I
+
=
W
¯
060
I
+
a
The reference sphere is at the intersection of the chief ray with the Gaussian image plane.
Table 11
Relationships between Intrinsic Coefficients
W
−
and
W
+
of a Spherical Surface
a
W
060
I
−
=
W
060
I
+
+
4
Ψ
W
040
·
W
¯
311
W
151
I
−
=
W
151
I
+
+
1
Ψ
[
3
W
131
·
W
¯
311
+
8
W
040
·
W
¯
220
+
8
W
040
·
W
¯
222
]
W
242
I
−
=
W
242
I
+
+
1
Ψ
[
2
W
222
·
W
¯
311
+
4
W
131
·
W
¯
220
+
6
W
131
·
W
¯
222
+
8
W
040
·
W
¯
131
]
W
333
I
−
=
W
333
I
+
+
1
Ψ
[
4
W
131
·
W
¯
131
+
4
W
222
·
W
¯
222
]
W
240
I
−
=
W
240
I
+
+
1
Ψ
[
2
W
220
·
W
¯
311
+
2
W
131
·
W
¯
220
+
4
W
040
·
W
¯
131
]
W
331
I
−
=
W
331
I
+
+
1
Ψ
[
5
W
131
·
W
¯
131
+
4
W
220
·
W
¯
222
+
4
W
222
·
W
¯
220
+
W
311
·
W
¯
311
+
16
W
040
·
W
¯
040
]
W
422
I
−
=
W
422
I
+
+
1
Ψ
[
6
W
222
·
W
¯
131
+
8
W
131
·
W
¯
040
+
2
W
311
·
W
¯
222
]
W
420
I
−
=
W
420
I
+
+
1
Ψ
[
2
W
220
·
W
¯
131
+
4
W
131
·
W
¯
040
]
W
511
I
−
=
W
511
I
+
+
1
Ψ
[
8
W
220
·
W
¯
040
+
8
W
222
·
W
¯
040
]
W
600
I
−
=
W
600
I
+
a
The reference sphere is centered at the intersection of the chief ray with the Gaussian image plane.
Table 12
Relationships between Sixth-Order Pupil and Image Aberration Coefficients for a Spherical Surface
a
W
¯
060
I
+
=
W
600
I
+
W
¯
151
I
+
=
W
511
I
+
−
3
8
Ψ
Δ
{
u
¯
4
}
+
1
Ψ
[
3
W
311
·
W
¯
131
+
8
W
220
·
W
¯
040
+
8
W
222
·
W
¯
040
]
−
3
2
W
311
u
¯
'
2
W
¯
242
I
+
=
W
422
I
+
−
3
4
Ψ
Δ
{
u
u
¯
3
}
+
1
Ψ
[
2
W
311
·
W
¯
222
+
4
W
220
·
W
¯
131
+
6
W
222
·
W
¯
131
+
8
W
131
·
W
¯
040
]
−
W
311
u
′
u
¯
'
W
¯
333
I
+
=
W
333
I
+
−
1
2
Ψ
Δ
{
u
2
u
¯
2
}
+
1
Ψ
[
4
W
131
·
W
¯
131
+
4
W
222
·
W
¯
222
]
W
¯
240
I
+
=
W
420
I
+
−
3
16
Ψ
Δ
{
u
u
¯
3
}
+
1
Ψ
[
2
W
311
·
W
¯
220
+
2
W
220
·
W
¯
131
+
4
W
131
·
W
¯
040
]
−
1
2
W
311
u
′
u
¯
'
W
¯
331
I
+
=
W
331
I
+
−
12
16
Ψ
Δ
{
u
2
u
¯
2
}
+
1
Ψ
[
5
W
131
·
W
¯
131
+
4
W
220
·
W
¯
220
+
4
W
220
·
W
¯
222
+
4
W
222
·
W
¯
220
+
W
311
·
W
¯
311
+
16
W
040
·
W
¯
040
]
+
1
2
W
¯
311
u
¯
'
2
−
1
2
W
311
u
'
2
W
¯
422
I
+
=
W
242
I
+
−
3
4
Ψ
Δ
{
u
3
u
¯
}
+
1
Ψ
[
2
W
222
·
W
¯
311
+
4
W
131
·
W
¯
220
+
6
W
131
·
W
¯
222
+
8
W
040
·
W
¯
131
]
+
W
¯
311
u
′
u
¯
'
W
¯
420
I
+
=
W
240
I
+
−
3
16
Ψ
Δ
{
u
3
u
¯
}
+
1
Ψ
[
2
W
220
·
W
¯
311
+
2
W
131
·
W
¯
220
+
4
W
040
·
W
¯
131
]
+
1
2
W
¯
311
u
′
u
¯
'
W
¯
511
I
+
=
W
151
I
+
−
3
8
Ψ
Δ
{
u
4
}
+
1
Ψ
[
3
W
131
·
W
¯
311
+
8
W
040
·
W
¯
220
+
8
W
040
·
W
¯
222
]
+
3
2
W
¯
311
u
'
2
W
¯
600
I
+
=
W
060
I
+
W
¯
331
I
+
=
W
331
I
+
−
12
16
Ψ
Δ
{
u
2
u
¯
2
}
+
1
Ψ
[
5
W
131
·
W
¯
131
+
4
W
220
·
W
¯
220
+
4
W
220
·
W
¯
222
+
4
W
222
·
W
¯
220
+
W
311
·
W
¯
311
+
16
W
040
·
W
¯
040
]
+
1
Ψ
W
311
[
W
131
−
W
¯
311
]
+
1
2
[
W
131
·
u
¯
'
2
−
W
¯
131
·
u
2
−
W
222
(
u
u
¯
+
u
′
u
¯
'
)
+
W
¯
222
(
u
u
¯
+
u
′
u
¯
'
)
]
a
The reference sphere is centered at the intersection of the chief ray with the Gaussian image plane.
Table 13
Sixth-Order Aberration Coefficients for a System of j Surfaces
W
060
+
=
∑
i
=
1
j
W
060
I
+
i
−
1
Ψ
∑
i
=
1
j
{
W
¯
311
i
∑
m
=
1
i
−
1
4
W
040
m
}
W
151
+
=
∑
i
=
1
j
W
151
I
+
i
−
1
Ψ
∑
i
=
1
j
{
W
¯
311
i
∑
m
=
1
i
−
1
3
W
131
m
}
−
1
Ψ
∑
i
=
1
j
{
W
¯
220
i
∑
m
=
1
i
−
1
8
W
040
m
}
−
1
Ψ
∑
i
=
1
j
{
W
¯
222
i
∑
m
=
1
i
−
1
8
W
040
m
}
W
242
+
=
∑
i
=
1
j
W
242
I
+
i
−
1
Ψ
∑
i
=
1
j
{
W
¯
311
i
∑
m
=
1
i
−
1
2
W
222
m
}
−
1
Ψ
∑
i
=
1
j
{
W
¯
220
i
∑
m
=
1
i
−
1
4
W
131
m
}
−
1
Ψ
∑
i
=
1
j
{
W
¯
222
i
∑
m
=
1
i
−
1
6
W
131
m
}
−
1
Ψ
∑
i
=
1
j
{
W
¯
131
i
∑
m
=
1
i
−
1
8
W
040
m
}
W
333
+
=
∑
i
=
1
j
W
333
I
+
i
−
1
Ψ
∑
i
=
1
j
{
W
¯
131
i
∑
m
=
1
i
−
1
4
W
131
m
}
−
1
Ψ
∑
i
=
1
j
{
W
¯
222
i
∑
m
=
1
i
−
1
4
W
222
m
}
W
240
+
=
∑
i
=
1
j
W
240
I
+
i
−
1
Ψ
∑
i
=
1
j
{
W
¯
220
i
∑
m
=
1
i
−
1
2
W
131
m
}
−
1
Ψ
∑
i
=
1
j
{
W
¯
311
i
∑
m
=
1
i
−
1
2
W
220
m
}
−
1
Ψ
∑
i
=
1
j
{
W
¯
131
i
∑
m
=
1
i
−
1
4
W
040
m
}
W
331
+
=
∑
i
=
1
j
W
331
I
+
i
−
1
Ψ
∑
i
=
1
j
{
W
¯
131
i
∑
m
=
1
i
−
1
5
W
131
m
}
−
1
Ψ
∑
i
=
1
j
{
W
¯
220
i
∑
m
=
1
i
−
1
4
W
220
m
}
−
1
Ψ
∑
i
=
1
j
{
W
¯
222
i
∑
m
=
1
i
−
1
4
W
220
m
}
−
1
Ψ
∑
i
=
1
j
{
W
¯
220
i
∑
m
=
1
i
−
1
4
W
222
m
}
−
1
Ψ
∑
i
=
1
j
{
W
¯
311
i
∑
m
=
1
i
−
1
W
311
m
}
−
1
Ψ
∑
i
=
1
j
{
W
¯
040
i
∑
m
=
1
i
−
1
16
W
040
m
}
+
∑
i
=
1
j
{
1
2
Δ
i
{
u
2
}
∑
m
=
1
i
−
1
W
311
m
}
W
422
+
=
∑
i
=
1
j
W
422
I
+
i
−
1
Ψ
∑
i
=
1
j
{
W
¯
222
i
∑
m
=
1
i
−
1
2
W
311
m
}
−
1
Ψ
∑
i
=
1
j
{
W
¯
131
i
∑
m
=
1
i
−
1
4
W
220
m
}
−
1
Ψ
∑
i
=
1
j
{
W
¯
131
i
∑
m
=
1
i
−
1
6
W
222
m
}
−
1
Ψ
∑
i
=
1
j
{
W
¯
040
i
∑
m
=
1
i
−
1
8
W
131
m
}
+
∑
i
=
1
j
{
Δ
i
{
u
u
¯
}
∑
m
=
1
i
−
1
W
311
m
}
W
420
+
=
∑
i
=
1
j
W
420
I
+
i
−
1
Ψ
∑
i
=
1
j
{
W
¯
131
i
∑
m
=
1
i
−
1
2
W
220
m
}
−
1
Ψ
∑
i
=
1
j
{
W
¯
220
i
∑
m
=
1
i
−
1
2
W
311
m
}
−
1
Ψ
∑
i
=
1
j
{
W
¯
040
i
∑
m
=
1
i
−
1
4
W
131
m
}
+
∑
i
=
1
j
{
1
2
Δ
i
{
u
u
¯
}
∑
m
=
1
i
−
1
W
311
m
}
W
511
+
=
∑
i
=
1
j
W
511
I
+
i
−
1
Ψ
∑
i
=
1
j
{
W
¯
131
i
∑
m
=
1
i
−
1
3
W
311
m
}
−
1
Ψ
∑
i
=
1
j
{
W
¯
040
i
∑
m
=
1
i
−
1
8
W
220
m
}
−
1
Ψ
∑
i
=
1
j
{
W
¯
040
i
∑
m
=
1
i
−
1
8
W
222
m
}
+
∑
i
=
1
j
{
3
2
Δ
i
{
u
¯
2
}
∑
m
=
1
i
−
1
W
311
m
}
W
600
+
=
∑
i
=
1
j
W
600
I
+
i
−
1
Ψ
∑
i
=
1
j
{
W
¯
040
i
∑
m
=
1
i
−
1
4
W
311
m
}
Table 14
Zero- and Second-Order Terms of the Irradiance Function
I
000
′
=
1
I
020
′
(
ρ
→
·
ρ
→
)
=
−
3
Ψ
W
¯
311
(
ρ
→
·
ρ
→
)
I
111
′
(
H
→
·
ρ
→
)
=
−
[
4
Ψ
W
¯
220
+
6
Ψ
W
¯
222
]
(
H
→
·
ρ
→
)
I
200
′
(
H
→
·
H
→
)
=
−
6
Ψ
W
¯
131
(
H
→
·
H
→
)
Table 15
Coefficient Comparison for an Aspheric Surface
a
W
151
−
0.4193360087176732
−
0.4193360087176732
W
242
1.3879576582604869 1.3879576582604876
W
333
−
1.3737719852861099
−
1.3737719852861099
W
240
0.1318037474153832 0.1318037474153833
W
331
−
1.5376298938401587
−
1.5376298938401600
W
422
0.4115466232465593 0.4115466232465597
W
420
2.3841052150066186 2.3841052150066186
W
511
−
2.2205687849607498
−
2.2205687849607463
a
The coefficients are in waves at
586.7
nm
.
Table 16
Sixth-Order Aberration Coefficients of the Triplet Lens in Waves at
587.6
nm
Sixth-Order Aberration Coefficients
W
040
W
131
W
222
W
220
W
311
3.095E-013 4.842E-013 7.749E-014
−
1.033
E
-
013
1.2695E-012
W
240
W
331
W
422
W
420
W
511
4.3519E-012
−
1.3257
E
-
013
4.2807E-014 3.4504E-014 8.6500E-013
W
060
W
151
W
242
W
333
1.011E-013 7.878E-014
−
1.176
E
-
012
−
8.605
E
-
014
Fifth-Order Transverse Aberration Coefficients (mm) B F C Pi E
−
6.71
E
-
15
−
2.522
E
-
015
−
3.537
E
-
016
1.414E-015
−
6.209
E
-
015
B5 F1 F2 M1 M2
−
3.5428
E
-
15
−
7.6113
E
-
017
−
5.187
E
-
017
1.2030E-014
−
9.6599
E
-
014
N1 N2 N3 C5 Pi5
−
2.536
E
-
016
−
4.7702
E
-
016
−
4.952
E
-
017
−
1.22
E
-
016
−
1.0157
E
-
016
M3 E5 1.252E-014
−
4.851
E
-
015
Table 17
Coefficients of the Sphere Function Difference
n
′
·
S
′
(
H
→
,
ρ
→
)
−
n
·
S
(
H
→
,
ρ
→
)
Second-order
n
′
S
020
′
−
n
S
020
=
−
1
2
Ψ
Δ
{
u
u
¯
}
n
′
S
111
′
−
n
S
111
=
−
Δ
{
Ψ
}
=
0
n
′
S
200
′
−
n
S
200
=
−
1
2
Ψ
Δ
{
u
¯
u
}
Fourth order
n
′
S
040
′
−
n
S
040
=
1
8
Ψ
Δ
{
u
3
u
¯
}
n
′
S
131
′
−
n
S
131
=
1
2
Ψ
Δ
{
u
2
}
n
′
S
222
′
−
n
S
222
=
1
2
Ψ
Δ
{
u
u
¯
}
n
′
S
220
′
−
n
S
220
=
1
4
Ψ
Δ
{
u
u
¯
}
n
′
S
311
′
−
n
S
311
=
1
2
Ψ
Δ
{
u
¯
2
}
n
′
S
400
′
−
n
S
400
=
1
8
Ψ
Δ
{
u
¯
3
u
}
Sixth order
n
′
S
060
′
−
n
S
060
=
−
1
16
Ψ
Δ
{
u
5
u
¯
}
n
′
S
151
'
−
n
S
151
=
−
3
8
Ψ
Δ
{
u
4
}
n
′
S
242
′
−
n
S
242
=
−
3
4
Ψ
Δ
{
u
3
u
¯
}
n
′
S
333
′
−
n
S
333
=
−
1
2
Ψ
Δ
{
u
2
u
¯
2
}
n
′
S
240
′
−
n
S
240
=
−
3
16
Ψ
Δ
{
u
3
u
¯
}
n
′
S
331
′
−
n
S
331
=
−
12
16
Ψ
Δ
{
u
2
u
¯
2
}
n
′
S
422
′
−
n
S
422
=
−
3
4
Ψ
Δ
{
u
u
¯
3
}
n
′
S
420
′
−
n
S
420
=
−
3
16
Ψ
Δ
{
u
u
¯
3
}
n
′
S
511
′
−
n
S
511
=
−
3
8
Ψ
Δ
{
u
¯
4
}
n
′
S
600
′
−
n
S
600
=
−
1
16
Ψ
Δ
{
u
¯
5
u
}
Table 18
Fourth-Order Aberrations Contributed by an Aspheric Cap where
A
4
is the Aspheric Coefficient
W
040
cap
=
Δ
{
n
}
·
A
4
·
y
4
W
¯
040
cap
=
Δ
{
n
}
·
A
4
·
y
¯
4
W
131
cap
=
4
Δ
{
n
}
·
A
4
·
y
3
·
y
¯
W
¯
131
cap
=
4
Δ
{
n
}
·
A
4
·
y
¯
3
·
y
W
222
cap
=
4
Δ
{
n
}
·
A
4
·
y
2
·
y
¯
2
W
¯
222
cap
=
4
Δ
{
n
}
·
A
4
·
y
¯
2
·
y
2
W
220
cap
=
2
Δ
{
n
}
·
A
4
·
y
2
·
y
¯
2
W
¯
220
cap
=
2
Δ
{
n
}
·
A
4
·
y
¯
2
·
y
2
W
311
cap
=
4
Δ
{
n
}
·
A
4
·
y
·
y
¯
3
W
¯
311
cap
=
4
Δ
{
n
}
·
A
4
·
y
¯
·
y
3
W
400
cap
=
Δ
{
n
}
·
A
4
·
y
¯
4
W
¯
400
cap
=
Δ
{
n
}
·
A
4
·
y
4
Table 19
Intrinsic Sixth-Order Aberrations Contributed by an Aspheric Cap
a
W
060
I
+
cap
=
−
1
2
Δ
{
n
·
u
2
}
·
A
4
·
y
4
+
2
·
W
040
cap
y
·
u
r
W
151
I
+
cap
=
u
y
n
′
Ψ
·
W
040
cap
+
3
Ψ
r
·
n
′
W
040
cap
+
8
Ψ
W
040
cap
·
W
220
P
W
242
I
+
cap
=
0
W
333
I
+
cap
=
0
W
240
I
+
cap
=
1
2
n
n
′
Ψ
2
·
W
040
cap
1
y
2
W
331
I
+
cap
=
0
W
422
I
+
cap
=
0
W
420
I
+
cap
=
0
W
511
I
+
cap
=
0
W
600
I
+
cap
=
0
a
The stop is located at the cap, and the aperture vector is located at the exit pupil.
Table 20
Intrinsic Sixth-Order Aberrations Contributed by an Aspheric Cap as the Stop is Shifted
a
W
060
I
+
=
W
060
I
+
cap
−
8
Ψ
y
¯
y
W
040
cap
·
W
040
cap
W
151
I
+
=
W
151
I
+
cap
+
6
y
¯
y
W
060
I
+
W
242
I
+
=
4
(
y
¯
y
)
W
151
I
+
cap
+
12
(
y
¯
y
)
2
W
060
I
+
W
333
I
+
=
4
(
y
¯
y
)
2
W
151
I
+
cap
+
8
(
y
¯
y
)
3
W
060
I
+
W
240
I
+
=
W
240
I
+
cap
+
(
y
¯
y
)
W
151
I
+
cap
+
3
(
y
¯
y
)
2
W
060
I
+
W
331
I
+
=
4
(
y
¯
y
)
W
240
I
+
cap
+
6
(
y
¯
y
)
2
W
151
I
+
cap
+
12
(
y
¯
y
)
3
W
060
I
+
+
1
Ψ
W
131
cap
·
W
311
cap
−
4
Ψ
(
y
¯
y
)
W
040
cap
·
W
311
cap
+
1
2
W
311
cap
·
u
·
u
W
422
I
+
=
4
(
y
¯
y
)
2
W
240
I
+
cap
+
8
(
y
¯
y
)
3
W
151
I
+
cap
+
12
(
y
¯
y
)
4
W
060
I
+
+
2
Ψ
W
222
cap
·
W
311
cap
−
8
Ψ
(
y
¯
y
)
2
W
040
cap
·
W
311
cap
+
W
311
cap
·
u
·
u
¯
W
420
I
+
=
2
(
y
¯
y
)
2
W
240
I
+
cap
+
2
(
y
¯
y
)
3
W
151
I
+
cap
+
3
(
y
¯
y
)
4
W
060
I
+
+
2
Ψ
W
220
cap
·
W
311
cap
−
4
Ψ
(
y
¯
y
)
2
W
040
cap
·
W
311
cap
+
1
2
W
311
cap
·
u
·
u
¯
W
511
I
+
=
4
(
y
¯
y
)
3
W
240
I
+
cap
+
5
(
y
¯
y
)
4
W
151
I
+
cap
+
6
(
y
¯
y
)
5
W
060
I
+
+
3
Ψ
W
311
cap
·
W
311
cap
−
12
Ψ
(
y
¯
y
)
3
W
040
cap
·
W
311
cap
+
3
2
W
311
cap
·
u
¯
·
u
¯
W
600
I
+
=
W
¯
060
I
+
a
The aperture vector is located at the exit pupil. The reference sphere is centered at the intersection of the chief ray with the Gaussian image plane.
Table 21
Sixth-Order Aberrations Contributed by an Aspheric Cap where
A
6
is the Sixth-Order Aspheric Coefficient
W
060
I
+
=
Δ
{
n
}
·
A
6
·
y
6
W
151
I
+
=
6
y
¯
y
W
060
I
+
W
242
I
+
=
12
(
y
¯
y
)
2
W
060
I
+
W
333
I
+
=
8
(
y
¯
y
)
3
W
060
I
+
W
240
I
+
=
3
(
y
¯
y
)
2
W
060
I
+
W
331
I
+
=
12
(
y
¯
y
)
3
W
060
I
+
W
422
I
+
=
12
(
y
¯
y
)
4
W
060
I
+
W
420
I
+
=
3
(
y
¯
y
)
4
W
060
I
+
W
511
I
+
=
6
(
y
¯
y
)
5
W
060
I
+
W
600
I
+
=
Δ
{
n
}
·
A
6
·
y
¯
6
Table 22
Quantities used in the Calculation of the Intrinsic Aberration Coefficients with the Aperture Vector
ρ
→
at the Exit Pupil
W
220
P
=
−
1
4
Ψ
2
P
W
240
C
C
+
=
+
1
16
A
r
Ψ
2
Δ
{
u
n
2
}
+
1
8
1
r
Ψ
2
Δ
{
u
2
n
}
+
1
4
y
2
r
2
W
220
P
+
y
r
u
′
W
220
P
−
1
4
u
r
Ψ
2
Δ
{
u
n
}
W
420
C
C
+
=
3
16
1
r
3
Ψ
4
Δ
{
1
n
}
1
A
2
(
W
420
C
C
+
=
0
for
A
=
0
)
W
331
C
C
+
=
−
2
W
220
P
·
u
′
u
¯
C
C
′
W
422
C
C
+
=
−
W
220
P
·
u
¯
C
C
′
2
W
151
C
C
+
=
−
4
W
040
·
u
′
u
¯
C
C
′
W
242
C
C
+
=
−
2
W
040
·
u
¯
C
C
′
2
u
¯
C
C
′
=
u
¯
′
−
A
¯
A
u
′
P
060
=
W
040
[
1
2
y
2
r
2
−
1
2
A
(
u
′
n
′
+
u
n
)
+
2
y
r
u
′
]
+
8
A
y
W
040
W
040
−
A
¯
A
8
Ψ
W
040
W
040
P
151
=
W
151
C
C
+
P
240
=
W
240
C
C
+
−
8
Ψ
A
¯
A
W
040
W
220
P
P
331
=
W
331
C
C
+
−
4
Ψ
A
¯
A
W
040
W
331
P
242
=
W
242
C
C
+
P
420
=
W
420
C
C
+
−
2
Ψ
A
¯
A
W
220
P
W
220
P
−
2
Ψ
W
220
P
W
311
P
511
=
−
2
Ψ
A
¯
A
W
220
P
W
311
C
331
=
1
Ψ
W
¯
311
W
311
+
1
2
W
311
u
u
C
422
=
2
Ψ
W
¯
222
W
311
+
W
311
u
u
¯
C
420
=
2
Ψ
W
¯
220
W
311
+
1
2
W
311
u
u
¯
C
511
=
3
Ψ
W
¯
131
W
311
+
3
2
W
311
u
¯
u
¯
Table 23
Intrinsic Aberration Coefficients for a Spherical Surface with the Aperture Vector
ρ
→
at the Exit Pupil
a
W
060
I
+
=
P
060
W
151
I
+
=
6
A
¯
A
P
060
+
P
151
W
242
I
+
=
12
(
A
¯
A
)
2
P
060
+
4
(
A
¯
A
)
P
151
+
P
242
W
333
I
+
=
8
(
A
¯
A
)
3
P
060
+
4
(
A
¯
A
)
2
P
151
+
2
(
A
¯
A
)
P
242
W
240
I
+
=
3
(
A
¯
A
)
2
P
060
+
(
A
¯
A
)
P
151
+
P
240
W
331
I
+
=
12
(
A
¯
A
)
3
P
060
+
6
(
A
¯
A
)
2
P
151
+
4
(
A
¯
A
)
P
240
+
2
(
A
¯
A
)
P
242
+
P
331
+
C
331
W
422
I
+
=
12
(
A
¯
A
)
4
P
060
+
8
(
A
¯
A
)
3
P
151
+
4
(
A
¯
A
)
2
P
240
+
5
(
A
¯
A
)
2
P
242
+
2
(
A
¯
A
)
P
331
+
P
422
+
C
422
W
420
I
+
=
3
(
A
¯
A
)
4
P
060
+
2
(
A
¯
A
)
3
P
151
+
2
(
A
¯
A
)
2
P
240
+
(
A
¯
A
)
2
P
242
+
(
A
¯
A
)
P
331
+
P
420
+
C
420
W
511
I
+
=
6
(
A
¯
A
)
5
P
060
+
5
(
A
¯
A
)
4
P
151
+
4
(
A
¯
A
)
3
P
240
+
4
(
A
¯
A
)
3
P
242
+
3
(
A
¯
A
)
2
P
331
+
2
(
A
¯
A
)
P
422
+
2
(
A
¯
A
)
P
420
+
P
511
+
C
511
W
600
I
+
=
W
¯
060
I
+
a
The reference sphere is centered at the intersection of the chief ray with the Gaussian image plane.
Table 24
Constructional Data (mm) of the Triplet Lens
a
Surface Object Radius Thickness 10,000 Glass
A
4
A
6
1 255.635318 56.8473 BK7
−
5.051563
e
-
07
−
3.2061469
e
-
11
2 62.646002 23.6149 3.5999265e-007 9.4325832e-010 3 74.494599 15.7912 BK7
−
1.0708598
e-6
1.151724e-010 4
−
58.717274
89.4891 5
−
42.87839
5 BK7 4.4688245e-007
−
3.256491
e
-
010
6
−
203.330401
4.2124 7
−
65.250745
Stop Image 65.249628
a
The exit pupil diameter is
14
mm
, and the field angle is 15°;
λ
=
587.6
nm
.