Weighted truncation of power series and the computation of chromatic aberration coefficients

G. W. Forbes

doi:10.1364/JOSAA.1.000350

Journal of the Optical Society of America A
Vol. 1,
Issue 4,
pp. 350-355
(1984)
•https://doi.org/10.1364/JOSAA.1.000350

Weighted truncation of power series and the computation of chromatic aberration coefficients

G. W. Forbes

Not Accessible

Your library or personal account may give you access

Get PDF
Email
Share
Get Citation
Copy Citation Text
G. W. Forbes, "Weighted truncation of power series and the computation of chromatic aberration coefficients," J. Opt. Soc. Am. A 1, 350-355 (1984)

Export Citation
- BibTex
- Endnote (RIS)
- HTML
- Plain Text
Citation alert
Save article

Abstract

In keeping with the recent growth of interest in the use of analytic computational methods in the analysis of the behavior of optical systems, this paper reports on the incorporation of chromatic dependence into these schemes. A novel prescription for the truncation of power series in several variables (which is also of value outside this context) is introduced in order to avoid much unnecessary computation. Subsequently, a process is devised for computing chromatic aberration coefficients to arbitrary orders for symmetric systems composed of homogeneous lenses and mirrors. It is found that, in less than the time required for the analytical analysis of a system at two distinct wavelengths, this process can yield the required results for the entire visible spectrum and beyond, if necessary.

Full Article | PDF Article

More Like This

Weighted order doubling in the computation of chromatic aberration coefficients

G. W. Forbes
J. Opt. Soc. Am. A 1(9) 974-980 (1984)

Chromatic coordinates in aberration theory

G. W. Forbes
J. Opt. Soc. Am. A 1(4) 344-349 (1984)

Order doubling in the computation of aberration coefficients

G. W. Forbes
J. Opt. Soc. Am. 73(6) 782-788 (1983)

Previous Article Next Article

Cited By

You do not have subscription access to this journal. Cited by links are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access Optica Member Subscription

Tables (4)

You do not have subscription access to this journal. Article tables are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access Optica Member Subscription

Equations (22)

You do not have subscription access to this journal. Equations are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access Optica Member Subscription

Weights	w₁	w₂	w₃	w₄	6	9	12
3 variables	1	1	1	∞	84	220	455
	1	2	3	∞	23	53	102
4 variables	1	1	1	1	210	715	1820
	1	2	3	3	31	84	186

				Time (sec)

Weights and Limits	Number of Coefficients	∊ (μm)	∊_p(μm)	Single Precision	Double Precision
(1, 1, 1, 5)	56	0.11	0.39	0.35	0.49
(1, 1, 1, 6)	84	0.05	0.18	0.62	0.90
(1, 1,1,7)	120	0.02	0.08	1.05	1.56
(2, 3, 4, 15)	54	0.04	0.13	0.37	0.51
(2, 3, 4, 16)	64	0.03	0.11	0.46	0.64
(1, 2, 3, 10)	67	0.04	0.12	0.49	0.70
(1, 2, 3, 11)	83	0.02	0.05	0.67	0.98

		∊(μm)			Time (sec)

Weights and Limits	Number of Coefficients	Range	rms	∊_p (μm)	Single Precision	Double Precision
(2, 3, 4, 4, 15)	95	0.03–0.08	(0.04)	0.14	0.63	0.90
(2, 2, 4, 4, 16)	119	0.03–0.04	(0.03)	0.11	0.84	1.22
(1, 2, 3, 3, 10)	111	0.03–0.05	(0.04)	0.12	0.80	1.16
(1, 2, 3, 3, 11)	144	0.01–0.02	(0.02	0.05	1.15	1.71

I	J	K	L	S	T	V	W
0	0	0	0	−0.18682086 − 003	0.12445631 + 003	−0.80310238 − 002	−0.26144047 + 001	1
1	0	0	0	0.44173130 − 006	−0.59635240 − 004	−0.13129058 − 007	0.10581862 − 005	2
0	1	0	0	−0.11924428 − 003	0.12048414 − 001	0.32426981 − 005	0.26440749 − 001	3
2	0	0	0	−0.28583982 − 009	−0.97608660 − 009	0.59394563 − 011	−0.30138253 − 009	4
0	0	0	1	0.53771663 − 003	0.38636271 − 001	−0.88811899 − 005	0.52456687 − 001	5
0	0	1	0	−0.27030948 − 002	0.20682435 + 003	0.13403705 − 001	−0.52145326 + 000	6
1	1	0	0	−0.38859233 − 008	0.10059849 − 004	0.82586094 − 009	0.38402983 − 006	7
3	0	0	0	0.10035194 − 012	0.14555514 − 011	−0.20147198 − 014	−0.11054952 − 012	8
1	0	0	1	−0.27127899 − 006	−0.19364801 − 005	0.57629253 − 008	0.89471551 − 006	9
1	0	1	0	−0.12161836 − 004	0.43982069 − 002	0.56594107 − 006	−0.18910665 − 003	10
0	2	0	0	0.85883085 − 005	−0.35537604 − 003	−0.22873508 − 006	−0.17022402 − 003	11
2	1	0	0	0.87015059 − 011	−0.48421051 − 009	−0.27224934 − 012	0.94142356 − 011	12
4	0	0	0	−0.14085947 − 016	−0.88644673 − 016	0.29053312 − 018	0.21652651 − 016	13
0	1	0	1	−0.39841615 − 005	0.16124171 − 002	0.25221399 − 006	−0.42747932 − 003	14
2	0	0	1	0.11713258 − 011	−0.15987720 − 009	−0.15513025 − 012	−0.39424409 − 010	15
0	1	1	0	0.91928881 − 002	−0.26303433 + 001	−0.36965521 − 003	0.33126103 − 001	16
2	0	1	0	−0.13733956 − 008	0.13771012 − 006	0.29153196 − 010	−0.11334952 − 007	17
1	2	0	0	0.94623258 − 009	−0.12243456 − 007	−0.22879825 − 010	−0.78665325 − 009	18
3	1	0	0	−0.63118741 − 015	0.32925916 − 012	0.54663045 − 016	0.34987264 − 014	19
5	0	0	0	−0.82188608 − 023	0.32966438 − 020	0.40198069 − 024	− 0,26842884 − 020	20
0	0	0	2	0.54318051 − 004	0.43695693 − 002	−0.63782761 − 006	0.54297074 − 002	21
0	0	1	1	0.12724232 − 001	−0.32195404 + 001	−0.46783224 − 003	−0.55462710 − 002	22
1	1	0	1	−0.61825682 − 009	−0.87272885 − 006	−0.10309882 − 009	−0.34427082 − 008	23
3	0	0	1	−0.36699448 − 014	−0.65641076 − 013	0.11392113 − 015	0.22555370 − 014	24
0	0	2	0	−0.71359735 + 000	0.59802506 + 003	0.31319251 − 001	−0.15823383 + 001	25
1	1	1	0	0.57572595 − 006	−0.60363159 − 004	−0.21277994 − 007	0.66702704 − 005	26
3	0	1	0	0.11807921 − 012	0.19348571 − 010	0.17773893 − 014	−0.73195026 − 012	27
0	3	0	0	0.42155567 − 009	0.26569003 − 004	0.37236895 − 008	0.67100468 − 006	28
2	2	0	0	0.50391932 − 014	0.41849550 − 010	0.47727469 − 014	0.42938249 − 013	29
4	1	0	0	−0.31681782 − 019	−0.39581472 − 016	−0.44144258 − 020	−0.50899768 − 018	30
6	0	0	0	0.51122010 − 026	−0.20302179 − 023	−0.32208776 − 027	−0.35275916 − 025	31
1	0	0	2	−0.28340956 − 007	−0.18799434 − 006	0.49055679 − 009	0.97349807 − 007	32
1	0	1	1	0.26227809 − 007	−0.54035329 − 004	−0.18574440 − 007	0.34602326 − 005	33
0	2	0	1	0.70327361 − 007	−0.34882971 − 004	−0.13034500 − 008	0.12768584 − 005	34
2	1	0	1	−0.31441897 − 012	−0.84981526 − 012	0.10051273 − 013	0.15501537 − 011	35
4	0	0	1	0.46038020 − 018	0.44375918 − 017	−0.14340595 − 019	−0.12325576 − 017	36
1	0	2	0	−0.13658749 − 003	0.42129454 − 001	0.42959942 − 005	−0.59129570 − 003	37
0	2	1	0	−0.55832484 − 004	0.43989049 − 001	0.64245356 − 005	−0.93770794 − 003	38
2	1	1	0	0.65143429 − 010	−0.71450338 − 008	−0.17198647 − 011	0.45932217 − 009	39
4	0	1	0	−0.21212636 − 016	−0.21741535 − 014	−0.16216848 − 018	−0.16863600 − 016	40
1	3	0	0	−0.12461164 − 011	0.33891352 − 008	0.50893782 − 012	−0.76329217 − 010	41
3	2	0	0	0.13214695 − 016	−0.55648915 − 014	−0.87735982 − 018	−0.13712775 − 016	42
5	1	0	0	−0.67020130 − 023	−0.10912580 − 020	−0.31003904 − 025	0.57628605 − 022	43
7	0	0	0	−0.37609619 − 030	0.21273422 − 027	0.52346076 − 031	0.38732126 − 030	44
0	1	0	2	−0.28470137 − 006	0.12217812 − 003	0.16733717 − 007	−0.48686977 − 004	45
2	0	0	2	0.10917640 − 012	−0.19802855 − 010	−0.17562991 − 013	−0.29789734 − 011	46
0	1	1	1	−0.10193517 − 003	0.60388451 − 001	0.12253387 − 004	0.34736410 − 003	47
2	0	1	1	−0.90491247 − 012	0.53233750 − 008	0.26598787 − 012	0.10364717 − 009	48
1	2	0	1	−0.39439441 − 010	0.86515798 − 008	0.21098526 − 011	−0.19488214 − 009	49
3	1	0	1	0.41327966 − 016	−0.17426849 − 013	−0.32097029 − 017	−0.50245785 − 016	50
5	0	0	1	−0.66727911 − 025	−0.94721807 − 021	−0.24742607 − 024	0.12300411 − 021	51
0	1	2	0	0.55277999 − 001	−0.21951176 + 002	−0.18882379 − 002	0.13145659 + 000	52
2	0	2	0	−0.12987315 − 007	0.22415217 − 005	0.33907885 − 009	−0.69459060 − 007	53
1	2	1	0	−0.15047599 − 008	0.18247845 − 005	0.39637959 − 009	−0.13651672 − 006	54
3	1	1	0	−0.12202027 − 014	0.10396871 − 011	−0.48532490 − 016	0.36204625 − 013	55
5	0	1	0	−0.13095325 − 020	0.23875873 − 018	0.45317612 − 022	−0.14759091 − 020	56
0	4	0	0	0.10253553 − 008	−0.39698043 − 006	−0.61168695 − 010	0.46381592 − 008	57
2	3	0,	0	0.12593479 − 014	−0.38885158 − 012	−0.62572442 − 016	−0.16319302 − 014	58
4	2	0	0	−0.19903099 − 020	0.40362659 − 018	0.71007016 − 022	0.63777650 − 020	59
6	1	0	0	0.71848002 − 027	0.55756122 − 025	0.22870415 − 029	0.72278480 − 027	60
8	0	0	0	0.32534086 − 034	−0.37356734 − 031	−0.72369619 − 035	−0.33374011 − 033	61
0	0	0	3	0.31700208 − 006	0.12452386 − 003	0.48150367 − 007	0.59440293 − 004	62
0	0	1	2	0.14309574 − 002	−0.36950374 + 000	−0.48404790 − 004	0.30004616 − 004	63
1	1	0	2	−0.11620786 − 009	−0.77939748 − 007	−0.83514339 − 011	−0.47525475 − 009	64
3	0	0	2	−0.38307573 − 015	−0.59949332 − 014	0.10427872 − 016	0.24491317 − 015	65
0	0	2	1	0.35545013 − 001	−0.14380848 + 002	−0.87828000 − 003	−0.23634920 − 001	66
1	1	1	1	0.59234930 − 008	−0.36409807 − 005	0.44063892 − 009	−0.38480089 − 007	67
3	0	1	1	−0.97337895 − 014	−0.31275614 − 012	0.15059984 − 016	−0.64785583 − 014	68
0	3	0	1	0.39936383 − 009	0.13589424 − 006	−0.14197927 − 011	0.20574871 − 008	69
2	2	0	1	0.53525097 − 015	−0.15707413 − 011	−0.18820425 − 015	−0.38124235 − 013	70
4	1	0	1	−0.17074504 − 020	0.17050915 − 017	0.20182389 − 021	0.33974562 − 019	71
6	0	0	1	−0.35922614 − 027	0.16214967 − 024	0.34830257 − 028	0.23609339 − 026	72
0	0	3	0	−0.33470255 + 001	0.20507714 + 004	0.89221261 − 001	−0.44196513 + 001	73
1	1	2	0	0.72466210 − 005	−0.19317466 − 002	−0.27071879 − 006	0.40209386 − 004	74
3	0	2	0	−0.50403219 − 012	0.20451370 − 009	0.28242318 − 013	−0.60461373 − 011	75
0	3	1	0	0.68893805 − 006	−0.44050688 − 003	−0.80898203 − 007	0.17367123 − 004	76
2	2	1	0	−0.10690962 − 011	0.46921351 − 009	0.54612302 − 013	−0.10472819 − 010	77
4	1	1	0	0.15429606 − 018	−0.10266026 − 015	0.14648804 − 020	0.45262062 − 018	78
6	0	1	0	−0.34433430 − 025	−0.60992110 − 024	0.22273468 − 026	−0.54234647 − 024	79
1	4	0	0	0.17164319 − 012	−0.49411680 − 010	−0.91056500 − 014	0.20531679 − 011	80
3	3	0	0	−0.14025768 − 018	0.94803725 − 016	0.14114965 − 019	−0.89877346 − 018	81
5	2	0	0	−0.17375856 − 026	−0.13671699 − 022	0.46719954 − 027	−0.11179490 − 023	82
7	1	0	0	−0.11220590 − 030	−0.18116918 − 028	−0.12817744 − 032	0.14856156 − 030	83
9	0	0	0	−0.88438896 − 038	0.19570677 − 035	0.52684524 − 039	0.46533527 − 037	84
1	0	0	3	−0.36498445 − 009	0.59211235 − 009	−0.15395327 − 010	0.20571410 − 008	85
1	0	1	2	0.18146606 − 007	−0.69093262 − 005	−0.18930958 − 008	0.42016907 − 006	86
0	2	0	2	0.50550853 − 008	−0.27008006 − 005	0.50695604 − 010	0.19180411 − 006	87
2	1	0	2	−0.34270793 − 013	0.29659794 − 012	0.10333714 − 014	0.13650359 − 012	88
4	0	0	2	0.48207189 − 019	0.36839462 − 018	−0.13337767 − 020	−0.11923167 − 018	89
1	0	2	1	0.17509402 − 005	−0.70671175 − 003	−0.12256193 − 006	0.25392956 − 005	90
0	2	1	1	0.26696710 − 005	−0.11973158 − 002	−0.22646567 − 006	−0.12968970 − 004	91
2	1	1	1	−0.11752935 − 012	−0.33235699 − 009	−0.16335586 − 013	0.60576793 − 011	92
4	0	1	1	0.60037703 − 018	0.15885029 − 015	0.25229526 − 019	−0.13883684 − 017	93
1	3	0	1	0.27576257 − 012	−0.21985953 − 009	−0.35666269 − 013	0.37143403 − 011	94
3	2	0	1	−0.97993926 − 018	0.32222513 − 015	0.55900329 − 019	−0.74837526 − 018	95
5	1	0	1	0.21622571 − 024	0.79616187 − 022	0.89497237 − 026	−0.22585769 − 023	96
7	0	0	1	0.37854401 − 031	−0.14341887 − 028	−0.37162221 − 032	−0.21551699 − 030	97
1	0	3	0	−0.79805110 − 003	0.27475663 +000	0.21033442 − 004	−0.20210620 − 002	98
0	2	2	0	−0.14499132 − 002	0.65078085 + 000	0.66363113 − 004	−0.57604526 − 002	99
2	1	2	0	0.77307393 − 009	−0.15213599 − 006	−0.25477600 − 010	0.48473111 − 008	100
4	0	2	0	−0.70361188 − 016	−0.86323491 − 014	0.40217717 − 018	−0.39692234 − 015	101
1	3	1	0	−0.73265448 − 011	0.35544811 − 008	−0.32137928 − 011	0.18616576 − 008	102
3	2	1	0	0.10457085 − 015	−0.39195373 − 013	−0.54519276 − 018	−0.89402298 − 015	103
5	1	1	0	0.33117949 − 022	−0.22577574 − 019	−0.25913796 − 023	0.22829898 − 021	104
7	0	1	0	−0.86721646 − 029	0.11579155 − 026	0.21979021 − 030	−0.32442112 − 028	105
0	5	0	0	−0.74446583 − 011	0.55085259 − 008	0.84694313 − 012	−0.16588539 − 009	106
2	4	0	0	0.55558468 − 017	0.53459023 − 014	0.65088920 − 018	0.95296720 − 017	107
4	3	0	0	0.75169215 − 023	−0.91566439 − 020	−0.10287452 − 023	−0.59967627 − 022	108
6	2	0	0	−0.26359094 − 028	0.24111574 − 028	0.12028383 − 030	0.42230483 − 029	109
8	1	0	0	0.38972143 − 035	0.14991126 − 032	0.18753230 − 036	−0.11437552 − 035	110
10	0	0	0	0.69250938 − 042	0.78752938 − 040	0.65402725 − 044	−0.44124016 − 041	111

Weights	w₁	w₂	w₃	w₄	6	9	12
3 variables	1	1	1	∞	84	220	455
	1	2	3	∞	23	53	102
4 variables	1	1	1	1	210	715	1820
	1	2	3	3	31	84	186

Abstract

Cited By

Tables (4)

Equations (22)

Journal of the Optical Society of America A