The continuing rapid increase in available computing power has not reduced the importance of efficient methods of optical system assessment for automatic lens design. On the contrary, the new capabilities simply show that truly automatic optical design will eventually be accomplished. It is proposed that the merit of a system-assessment scheme be measured in terms of the accuracy of its estimation of the overall performance of a proposed system as a function of the amount of work done (e.g., number of rays traced). By using this criterion, a number of schemes based on ray tracing are compared, and some highly efficient assessment procedures are developed. As a simplifying approximation, the effects of vignetting and pupil distortion are ignored here. The key to the most-effective methods lies in coupling appropriate coordinates to Gaussian quadrature schemes. Appropriate coordinate systems are those for which the relevant integrands (either wave-front errors or transverse intercept errors) take the form of smooth functions. The resulting methods for system assessment are typically at least an order of magnitude more efficient than comparatively simple schemes.
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Gaussian Integration Parameters for the Radial Integral as Approximated in Eq. (3.5)
Nr
j
ρj1/2
wj
1
1
0.70710678
0.50000000
2
1
0.45970084
0.25000000
2
0.88807383
0.25000000
3
1
0.33571069
0.13888889
2
0.70710678
0.22222222
3
0.94196515
0.13888889
4
1
0.26349923
0.08696371
2
0.57446451
0.16303629
3
0.81852949
0.16303629
4
0.96465961
0.08696371
5
1
0.21658734
0.05923172
2
0.48038042
0.11965717
3
0.70710678
0.14222222
4
0.87706023
0.11965717
5
0.97626324
0.05923172
6
1
0.18375321
0.04283112
2
0.41157661
0.09019039
3
0.61700114
0.11697848
4
0.78696226
0.11697848
5
0.91137517
0.09019039
6
0.98297241
0.04283112
Table 2
Radau Integration Parameters for the Radial Integral as Approximated in Eq. (3.6)
Nr
j
uj
υj
1
0
0.00000000
0.12500000
1
0.81649658
0.37500000
2
0
0.00000000
0.05555556
1
0.59586158
0.25624291
2
0.91921106
0.18820153
3
0
0.00000000
0.03125000
1
0.46080423
0.16442216
2
0.76846154
0.19409673
3
0.95467902
0.11023111
4
0
0.00000000
0.02000000
1
0.37384471
0.11155195
2
0.64529805
0.15591326
3
0.85038637
0.14067801
4
0.97102822
0.07185678
5
0
0.00000000
0.01388889
1
0.31390299
0.07991019
2
0.55184756
0.12134680
3
0.74968339
0.13023170
4
0.89553704
0.10422533
5
0.97989292
0.05039710
Table 3
Gaussian Integration Parameters for the Radial Integral in Eq. (3.8) with S = 0.2 (i.e., U ≈ 27°)
Nr
j
sj
xj
1
1
0.69384445
0.50000000
2
1
0.43543723
0.23070844
2
0.87510456
0.26929156
3
1
0.31232501
0.12232506
2
0.67840051
0.22109640
3
0.93304442
0.15657855
4
1
0.24273576
0.07465129
2
0.54098777
0.15278777
3
0.79417534
0.17191291
4
0.95844762
0.10064803
5
1
0.19831273
0.05006067
2
0.44698355
0.10783627
3
0.67414583
0.14117340
4
0.85757794
0.13124438
5
0.97175498
0.06968528
6
1
0.16756660
0.03583065
2
0.37990375
0.07919620
3
0.58067807
0.11097140
4
0.75762209
0.12141973
5
0.89583930
0.10163465
6
0.97957228
0.05094737
Table 4
Gaussian Integration Parameters for the Radial Integral in Eq. (3.8) with S = 0.4 (i.e., U ≈ 39°)
Nr
j
sj
xj
1
1
0.67647261
0.50000000
2
1
0.40432614
0.20615699
2
0.85723961
0.29384301
3
1
0.28334434
0.10279606
2
0.64005104
0.21639410
3
0.92016371
0.18080984
4
1
0.21754825
0.06076048
2
0.49792986
0.13813168
3
0.76021048
0.18081687
4
0.94919652
0.12029096
5
1
0.17644476
0.03998941
2
0.40520854
0.09278801
3
0.62998719
0.13682736
4
0.82948791
0.14523767
5
0.96490002
0.08515755
6
1
0.14837563
0.02827855
2
0.34105017
0.06602071
3
0.53356700
0.10141885
4
0.71692624
0.12446318
5
0.87287672
0.11660231
6
0.97432582
0.06321641
Table 5
Gaussian Integration Parameters for the Radial Integral in Eq. (3.8) with S = 0.6 (i.e., U ≈ 51°)
Nr
j
sj
xj
1
1
0.65168244
0.50000000
2
1
0.36130432
0.17288555
2
0.83010332
0.32711445
3
1
0.24509622
0.07896118
2
0.58426612
0.20408175
3
0.89935733
0.21695707
4
1
0.18520696
0.04473904
2
0.43851128
0.11612775
3
0.70803535
0.18758867
4
0.93360380
0.15154454
5
1
0.14883717
0.02875378
2
0.34963074
0.07296355
3
0.56590348
0.12571259
4
0.78438867
0.16168849
5
0.95299762
0.11088159
6
1
0.12441841
0.02003190
2
0.29061007
0.04984692
3
0.46811830
0.08585576
4
0.65517275
0.12322550
5
0.83472366
0.13677914
6
0.96501817
0.08426077
Table 6
Gaussian Integration Parameters for the Radial Integral in Eq. (3.8) with S = 0.8 (i.e., U ≈ 63°)
Nr
j
sj
xj
1
1
0.60937257
0.50000000
2
1
0.29185413
0.12205389
2
0.77964770
0.37794611
3
1
0.18770381
0.04804410
2
0.48778717
0.17142335
3
0.85701314
0.28053255
4
1
0.13853492
0.02554845
2
0.34408819
0.07987852
3
0.61061112
0.18218416
4
0.89966525
0.21238888
5
1
0.10989681
0.01587943
2
0.26589927
0.04527436
3
0.45695849
0.09808976
4
0.69444529
0.17597196
5
0.92577314
0.16478450
6
1
0.09112174
0.01084000
2
0.21711832
0.02919158
3
0.36395062
0.05872956
4
0.54335308
0.10759254
5
0.75440686
0.16281059
6
0.94291541
0.13083573
Table 7
Gaussian Integration Parameters for Integration over the Visible Spectrum with Uniform Weight in Wavelength and with the Gaussian Weight Plotted in Fig. 9a
Nr
j
Value of Integration Parameter (μm)
Uniform Weight
Gaussian Weight
λj
wj
λj
wj
1
1
0.528944
0.300000
0.541368
0.1433636
2
1
0.440626
0.106793
0.471604
0.0443355
2
0.608925
0.193208
0.584579
0.0990281
3
1
0.418886
0.049456
0.434658
0.0099821
2
0.505546
0.123853
0.518983
0.0782096
3
0.644536
0.126692
0.614795
0.0551719
4
1
0.410836
0.028202
0.418309
0.0032488
2
0.460136
0.071776
0.477524
0.0316589
3
0.554033
0.112203
0.552481
0.0795135
4
0.662979
0.087820
0.637241
0.0289423
It is emphasised that these parameters were determined by using Buchdahl’s chromatic coordinate as the underlying variable, which entails that the weight function pick up a Jacobian.
Table 8
Gaussian Integration Parameters for Averaging over the Field by Using the Weight Function Presented in Fig. 12
Nr
j
sj
xj
1
1
0.64168895
0.35416667
2
1
0.43031871
0.20629209
2
0.85315256
0.14787458
3
1
0.31999532
0.12240883
2
0.67815443
0.16406851
3
0.92442742
0.06768933
4
1
0.25381393
0.07923447
2
0.55436991
0.13417354
3
0.79663522
0.10415890
4
0.95517022
0.03659975
5
1
0.21003785
0.05503557
2
0.46615087
0.10422657
3
0.68857948
0.10580761
4
0.86125653
0.06664974
5
0.97070391
0.02244717
6
1
0.17903360
0.04031148
2
0.40109978
0.08121040
3
0.60226192
0.09485103
4
0.77138936
0.07809421
5
0.89995791
0.04464430
6
0.97948621
0.01505525
Table 9
Radau Integration Parameters for Averaging over the Field by Using the Weight Function Presented in Fig. 12
Nr
j
uj
υj
1
0
0.00000000
0.10518293
1
0.76531973
0.24898374
2
0
0.00000000
0.04958584
1
0.56484166
0.20324952
2
0.89399859
0.10133130
3
0
0.00000000
0.02870831
1
0.44198632
0.14150922
2
0.74279304
0.13411331
3
0.94168944
0.04983582
4
0
0.00000000
0.01868944
1
0.36144719
0.10008973
2
0.62563117
0.12268764
3
0.83161892
0.08406164
4
0.96374493
0.02863823
5
0
0.00000000
0.01312624
1
0.30516908
0.07346590
2
0.53709792
0.10245377
3
0.73253887
0.09202967
4
0.88203922
0.05472700
5
0.97548572
0.01836408
Tables (9)
Table 1
Gaussian Integration Parameters for the Radial Integral as Approximated in Eq. (3.5)
Nr
j
ρj1/2
wj
1
1
0.70710678
0.50000000
2
1
0.45970084
0.25000000
2
0.88807383
0.25000000
3
1
0.33571069
0.13888889
2
0.70710678
0.22222222
3
0.94196515
0.13888889
4
1
0.26349923
0.08696371
2
0.57446451
0.16303629
3
0.81852949
0.16303629
4
0.96465961
0.08696371
5
1
0.21658734
0.05923172
2
0.48038042
0.11965717
3
0.70710678
0.14222222
4
0.87706023
0.11965717
5
0.97626324
0.05923172
6
1
0.18375321
0.04283112
2
0.41157661
0.09019039
3
0.61700114
0.11697848
4
0.78696226
0.11697848
5
0.91137517
0.09019039
6
0.98297241
0.04283112
Table 2
Radau Integration Parameters for the Radial Integral as Approximated in Eq. (3.6)
Nr
j
uj
υj
1
0
0.00000000
0.12500000
1
0.81649658
0.37500000
2
0
0.00000000
0.05555556
1
0.59586158
0.25624291
2
0.91921106
0.18820153
3
0
0.00000000
0.03125000
1
0.46080423
0.16442216
2
0.76846154
0.19409673
3
0.95467902
0.11023111
4
0
0.00000000
0.02000000
1
0.37384471
0.11155195
2
0.64529805
0.15591326
3
0.85038637
0.14067801
4
0.97102822
0.07185678
5
0
0.00000000
0.01388889
1
0.31390299
0.07991019
2
0.55184756
0.12134680
3
0.74968339
0.13023170
4
0.89553704
0.10422533
5
0.97989292
0.05039710
Table 3
Gaussian Integration Parameters for the Radial Integral in Eq. (3.8) with S = 0.2 (i.e., U ≈ 27°)
Nr
j
sj
xj
1
1
0.69384445
0.50000000
2
1
0.43543723
0.23070844
2
0.87510456
0.26929156
3
1
0.31232501
0.12232506
2
0.67840051
0.22109640
3
0.93304442
0.15657855
4
1
0.24273576
0.07465129
2
0.54098777
0.15278777
3
0.79417534
0.17191291
4
0.95844762
0.10064803
5
1
0.19831273
0.05006067
2
0.44698355
0.10783627
3
0.67414583
0.14117340
4
0.85757794
0.13124438
5
0.97175498
0.06968528
6
1
0.16756660
0.03583065
2
0.37990375
0.07919620
3
0.58067807
0.11097140
4
0.75762209
0.12141973
5
0.89583930
0.10163465
6
0.97957228
0.05094737
Table 4
Gaussian Integration Parameters for the Radial Integral in Eq. (3.8) with S = 0.4 (i.e., U ≈ 39°)
Nr
j
sj
xj
1
1
0.67647261
0.50000000
2
1
0.40432614
0.20615699
2
0.85723961
0.29384301
3
1
0.28334434
0.10279606
2
0.64005104
0.21639410
3
0.92016371
0.18080984
4
1
0.21754825
0.06076048
2
0.49792986
0.13813168
3
0.76021048
0.18081687
4
0.94919652
0.12029096
5
1
0.17644476
0.03998941
2
0.40520854
0.09278801
3
0.62998719
0.13682736
4
0.82948791
0.14523767
5
0.96490002
0.08515755
6
1
0.14837563
0.02827855
2
0.34105017
0.06602071
3
0.53356700
0.10141885
4
0.71692624
0.12446318
5
0.87287672
0.11660231
6
0.97432582
0.06321641
Table 5
Gaussian Integration Parameters for the Radial Integral in Eq. (3.8) with S = 0.6 (i.e., U ≈ 51°)
Nr
j
sj
xj
1
1
0.65168244
0.50000000
2
1
0.36130432
0.17288555
2
0.83010332
0.32711445
3
1
0.24509622
0.07896118
2
0.58426612
0.20408175
3
0.89935733
0.21695707
4
1
0.18520696
0.04473904
2
0.43851128
0.11612775
3
0.70803535
0.18758867
4
0.93360380
0.15154454
5
1
0.14883717
0.02875378
2
0.34963074
0.07296355
3
0.56590348
0.12571259
4
0.78438867
0.16168849
5
0.95299762
0.11088159
6
1
0.12441841
0.02003190
2
0.29061007
0.04984692
3
0.46811830
0.08585576
4
0.65517275
0.12322550
5
0.83472366
0.13677914
6
0.96501817
0.08426077
Table 6
Gaussian Integration Parameters for the Radial Integral in Eq. (3.8) with S = 0.8 (i.e., U ≈ 63°)
Nr
j
sj
xj
1
1
0.60937257
0.50000000
2
1
0.29185413
0.12205389
2
0.77964770
0.37794611
3
1
0.18770381
0.04804410
2
0.48778717
0.17142335
3
0.85701314
0.28053255
4
1
0.13853492
0.02554845
2
0.34408819
0.07987852
3
0.61061112
0.18218416
4
0.89966525
0.21238888
5
1
0.10989681
0.01587943
2
0.26589927
0.04527436
3
0.45695849
0.09808976
4
0.69444529
0.17597196
5
0.92577314
0.16478450
6
1
0.09112174
0.01084000
2
0.21711832
0.02919158
3
0.36395062
0.05872956
4
0.54335308
0.10759254
5
0.75440686
0.16281059
6
0.94291541
0.13083573
Table 7
Gaussian Integration Parameters for Integration over the Visible Spectrum with Uniform Weight in Wavelength and with the Gaussian Weight Plotted in Fig. 9a
Nr
j
Value of Integration Parameter (μm)
Uniform Weight
Gaussian Weight
λj
wj
λj
wj
1
1
0.528944
0.300000
0.541368
0.1433636
2
1
0.440626
0.106793
0.471604
0.0443355
2
0.608925
0.193208
0.584579
0.0990281
3
1
0.418886
0.049456
0.434658
0.0099821
2
0.505546
0.123853
0.518983
0.0782096
3
0.644536
0.126692
0.614795
0.0551719
4
1
0.410836
0.028202
0.418309
0.0032488
2
0.460136
0.071776
0.477524
0.0316589
3
0.554033
0.112203
0.552481
0.0795135
4
0.662979
0.087820
0.637241
0.0289423
It is emphasised that these parameters were determined by using Buchdahl’s chromatic coordinate as the underlying variable, which entails that the weight function pick up a Jacobian.
Table 8
Gaussian Integration Parameters for Averaging over the Field by Using the Weight Function Presented in Fig. 12
Nr
j
sj
xj
1
1
0.64168895
0.35416667
2
1
0.43031871
0.20629209
2
0.85315256
0.14787458
3
1
0.31999532
0.12240883
2
0.67815443
0.16406851
3
0.92442742
0.06768933
4
1
0.25381393
0.07923447
2
0.55436991
0.13417354
3
0.79663522
0.10415890
4
0.95517022
0.03659975
5
1
0.21003785
0.05503557
2
0.46615087
0.10422657
3
0.68857948
0.10580761
4
0.86125653
0.06664974
5
0.97070391
0.02244717
6
1
0.17903360
0.04031148
2
0.40109978
0.08121040
3
0.60226192
0.09485103
4
0.77138936
0.07809421
5
0.89995791
0.04464430
6
0.97948621
0.01505525
Table 9
Radau Integration Parameters for Averaging over the Field by Using the Weight Function Presented in Fig. 12