When designing an optical system it is generally accepted that it
is too costly to generate the entire Hessian for every iteration of an
optimizer. However, the Hessian also is useful in tolerance
analysis for which it needs to be calculated only once. We propose
using the Hessian as part of a cost-based tolerancing
procedure. Considerations for the general implementation of the
proposed ideas are discussed, and the utility of this approach is
demonstrated by way of an example. In the example optimal
manufacturing tolerances are determined for a doublet. As expected,
the optimal tolerances change as quantities such as the requisite image
quality for finished systems, manufacturing yields, and relative
expenses of meeting given tolerances are varied.
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The object is located at infinity, and
the stop is in contact with the first surface.
The Gaussian image height.
Table 2
Gradient of the Uncompensated Example
System: (∂M/∂si
) is
Given for the si
That Appear in the First Row
R
1
R
2
R
3
t
1
t
2
α
β
δy
tim
-6.13 × 10-5
-6.06 × 10-5
1.29 × 10-5
-1.06 × 10-5
-5.33 × 10-5
0.0
0.0
0.0
1.00 × 10-9
Table 3
Hessian of the Uncompensated Example System with the
Elements
(∂2M/∂si
∂sj
)
a
si
sj
R
1
R
2
R
3
t
1
t
2
α
β
δy
tim
R
1
23.7589
10.8230
-4.04579
-8.35652
-3.76589
0.0
0.0
0.0
-16.9851
R
2
10.8230
4.93145
-1.84307
-3.80673
-1.71523
0.0
0.0
0.0
-7.73689
R
3
-4.04579
-1.84307
0.688945
1.42300
0.641255
0.0
0.0
0.0
2.89228
t
1
-8.35652
-3.80673
1.42300
2.93918
1.32453
0.0
0.0
0.0
5.97401
t
2
-3.76589
-1.71523
0.641255
1.32453
0.596970
0.0
0.0
0.0
2.69228
α
0.0
0.0
0.0
0.0
0.0
0.020367
0.0
0.013353
0.0
β
0.0
0.0
0.0
0.0
0.0
0.0
0.020367
0.0
0.0
δy
0.0
0.0
0.0
0.0
0.0
0.013353
0.0
0.134574
0.0
tim
-16.9851
-7.73689
2.89228
5.97401
2.69228
0.0
0.0
0.0
12.1426
si and
sj are specified by the associated row and
column headings.
Table 4
Parameters That Specify the Individual Expense
Functionsa
Variable
Δsa
Δsb
Δsc
Δsd
Radii of curvature (fringes)
6
15
30
50
R
1
(mm)
0.125
0.312
0.621
1.028
R
2
(mm)
0.053
0.132
0.265
0.443
R
3
(mm)
0.829
2.089
4.234
7.184
t
1
(mm)
0.004
0.01
0.02
0.04
t
2
(mm)
0.004
0.01
0.02
0.04
Tilt (deg)
0.04
0.1
0.2
0.4
δy (mm)
0.004
0.01
0.02
0.04
Note that the fringes are taken to be
double pass and evaluated at λ = 546.1 nm. The corresponding
values (in millimeters) for the three radii are listed separately.
Table 5
Parameters for the Five Different Cases of the
Examplea
Case
Mmax
Rmax (%)
δc for R
1
and Tilt
1
1.05
M
0
1.0
3.0
2
1.05
M
0
1.0
1.0
3
1.05
M
0
5.0
1.0
4
1.20
M
0
1.0
1.0
5
1.20
M
0
5.0
1.0
The cases are distinguished by the
maximum acceptable figure of merit Mmax, the
maximum rejection rate Rmax, and the value of
δc for R1 and the tilt of the
entire doublet.
Table 6
Resultant Optimal Tolerances for the Given Variables after
Optimization of the Five Different Casesa
CaseΔR
1
(fringes)
ΔR
1
(fringes)
ΔR
2
(fringes)
ΔR
3
(fringes)
Δt
1
(mm)
Δt
2
(mm)
ΔTilt (deg)
Δδy (mm)
Relative Cost
1
15.3
6.9
6.4
0.0337
0.0398
0.0574
0.0203
14.65
2
9.6
15.4
6.2
0.0383
0.0400
0.0500
0.0205
7.89
3
18.5
17.1
7.9
0.0221
0.0255
0.0450
0.0201
7.19
4
32.4
17.8
16.0
0.0397
0.0400
0.1079
0.0399
3.72
5
45.3
30.3
18.7
0.0244
0.0314
0.1035
0.0273
2.97
The last column gives the relative cost
to produce systems with the corresponding tolerances. The fringes
are double pass and evaluated at 546.1 nm.
The object is located at infinity, and
the stop is in contact with the first surface.
The Gaussian image height.
Table 2
Gradient of the Uncompensated Example
System: (∂M/∂si
) is
Given for the si
That Appear in the First Row
R
1
R
2
R
3
t
1
t
2
α
β
δy
tim
-6.13 × 10-5
-6.06 × 10-5
1.29 × 10-5
-1.06 × 10-5
-5.33 × 10-5
0.0
0.0
0.0
1.00 × 10-9
Table 3
Hessian of the Uncompensated Example System with the
Elements
(∂2M/∂si
∂sj
)
a
si
sj
R
1
R
2
R
3
t
1
t
2
α
β
δy
tim
R
1
23.7589
10.8230
-4.04579
-8.35652
-3.76589
0.0
0.0
0.0
-16.9851
R
2
10.8230
4.93145
-1.84307
-3.80673
-1.71523
0.0
0.0
0.0
-7.73689
R
3
-4.04579
-1.84307
0.688945
1.42300
0.641255
0.0
0.0
0.0
2.89228
t
1
-8.35652
-3.80673
1.42300
2.93918
1.32453
0.0
0.0
0.0
5.97401
t
2
-3.76589
-1.71523
0.641255
1.32453
0.596970
0.0
0.0
0.0
2.69228
α
0.0
0.0
0.0
0.0
0.0
0.020367
0.0
0.013353
0.0
β
0.0
0.0
0.0
0.0
0.0
0.0
0.020367
0.0
0.0
δy
0.0
0.0
0.0
0.0
0.0
0.013353
0.0
0.134574
0.0
tim
-16.9851
-7.73689
2.89228
5.97401
2.69228
0.0
0.0
0.0
12.1426
si and
sj are specified by the associated row and
column headings.
Table 4
Parameters That Specify the Individual Expense
Functionsa
Variable
Δsa
Δsb
Δsc
Δsd
Radii of curvature (fringes)
6
15
30
50
R
1
(mm)
0.125
0.312
0.621
1.028
R
2
(mm)
0.053
0.132
0.265
0.443
R
3
(mm)
0.829
2.089
4.234
7.184
t
1
(mm)
0.004
0.01
0.02
0.04
t
2
(mm)
0.004
0.01
0.02
0.04
Tilt (deg)
0.04
0.1
0.2
0.4
δy (mm)
0.004
0.01
0.02
0.04
Note that the fringes are taken to be
double pass and evaluated at λ = 546.1 nm. The corresponding
values (in millimeters) for the three radii are listed separately.
Table 5
Parameters for the Five Different Cases of the
Examplea
Case
Mmax
Rmax (%)
δc for R
1
and Tilt
1
1.05
M
0
1.0
3.0
2
1.05
M
0
1.0
1.0
3
1.05
M
0
5.0
1.0
4
1.20
M
0
1.0
1.0
5
1.20
M
0
5.0
1.0
The cases are distinguished by the
maximum acceptable figure of merit Mmax, the
maximum rejection rate Rmax, and the value of
δc for R1 and the tilt of the
entire doublet.
Table 6
Resultant Optimal Tolerances for the Given Variables after
Optimization of the Five Different Casesa
CaseΔR
1
(fringes)
ΔR
1
(fringes)
ΔR
2
(fringes)
ΔR
3
(fringes)
Δt
1
(mm)
Δt
2
(mm)
ΔTilt (deg)
Δδy (mm)
Relative Cost
1
15.3
6.9
6.4
0.0337
0.0398
0.0574
0.0203
14.65
2
9.6
15.4
6.2
0.0383
0.0400
0.0500
0.0205
7.89
3
18.5
17.1
7.9
0.0221
0.0255
0.0450
0.0201
7.19
4
32.4
17.8
16.0
0.0397
0.0400
0.1079
0.0399
3.72
5
45.3
30.3
18.7
0.0244
0.0314
0.1035
0.0273
2.97
The last column gives the relative cost
to produce systems with the corresponding tolerances. The fringes
are double pass and evaluated at 546.1 nm.