Improved solution for the cemented doublet

Abraham Szulc

doi:10.1364/AO.35.003548

Applied Optics
Vol. 35,
Issue 19,
pp. 3548-3558
(1996)
•https://doi.org/10.1364/AO.35.003548

Improved solution for the cemented doublet

Abraham Szulc

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Abraham Szulc, "Improved solution for the cemented doublet," Appl. Opt. 35, 3548-3558 (1996)

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Abstract

A method is described that permits the calculation of a cemented doublet with a given spherical aberration and coma at the edge of the lens. In particular the aberrations can be set to zero. Given one glass, the equations reported in this paper permit the determination of a second matching glass that minimizes the spherochromatism and coma of the lens. This result is obtained by the introduction, into the third-order thin-lens formulas, the third-order values of the aberration coefficients, as derived from the equation developed by Mossotti which yields zero finite aberrations for the same lens with added thickness. After a brief historical introduction, the third-order equations are developed and tables for the color-correcting glasses and S _I and S _II (the Seidel third-order coefficients) are given for objects at infinity and at a magnification of −1, both for flint- and crown-leading cases. The paper closes with a table of corrected doublets.

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Aberration (units)	Rayleigh Tolerance	F/3	F/4	F/5
RAN (μm)	2λ(F/D)²	10.4	18.6	29.0
PSP (μm)	16λ(F/D)²	83	149	232
ZSP (μm)	24λ(F/D)²	125	223	350
PCR (μm)	4λ(F/D)²	20.8	37.2	58.0
SCR (μm)	5λ(F/D)²	26.0	46.5	75.5
SCR (λ)	dw′	0.62	0.62	0.62
CVS (λ)	Δdw′	2.0	2.0	2.0
OSC′ (10⁻⁴)	$1.8 λ (F / D) / y_{k}^{'}$	6.3	8.4	10.0

Aberration	F/D			Units

	F/3	F/4	F/5
$Δ S_{m}^{'}$	0.975	0.265	0.102	Millimeters
$Δ S_{z}^{'}$	0.205	0.061	0.024	Millimeters
PCR	0.000	0.000	0.000	Millimeters
SCR	−4.84	−1.40	−0.51	λ _d
CVS	−4.87	−1.40	−0.55	λ _d
OSC′	−21.2	−5.65	−2.15	×10⁻⁴

Variable	F/D			Units

	F/3	F/4	F/5
$Δ S_{m}^{'}$	0.000	0.000	0.000	Millimeters
$Δ S_{z}^{'}$	−0.18	−0.06	−0.02	Millimeters
PCR	−0.15	−0.08	−0.05	Millimeters
SCR	−0.12	−0.11	−0.07	λ _d
CVS	−3.60	−1.18	−0.49	λ _d
OSC′	3.74	1.05	0.42	×10⁻⁴

F/D	Glass	R	T
F/3	517642	62.40	8.00
	648331	−43.25	3.00
		−132.77	94.65
F/4	517642	61.79	6.00
	648331	−42.49	3.00
		−134.20	95.71
F/5	517642	61.68	4.00
	648331	−42.37	3.00
		−134.93	96.82

Flint Leading						Crown Leading						Flint Leading m = −1

m = 0						m = 0

n ₁, ν₁	n ₂	ν₂	Δν₂			n ₁	ν₁	n₂, ν₂	Δν₁			n ₁, ν₁	n ₂	ν₂	Δν2

			F/3	F/4	F/5				F/3	F/4	F/5				F/3	F/4	F/5
1.88, 22.70	1.48	47.32	1.93	1.96	2.17	1.48	50.49	1.88, 22.70	0.20	0.32	0.46	1.88, 22.70	1.48	48.58	0.91	0.97	1.14
	1.56	44.94	1.21	1.16	1.34	1.56	47.90		−0.01	0.08	0.17		1.56	46.02	0.49	0.52	0.62
	1.64	42.41	0.74	0.73	0.81	1.64	44.87		−0.03	0.06	0.12		1.64	43.21	0.26	0.27	0.34
	1.72	39.45	0.44	0.43	0.49	1.72	41.34		−0.03	0.03	0.08		1.72	39.95	0.12	0.13	0.18
	1.80	35.54	0.26	0.25	0.31	1.80	36.68		0.00	0.05	0.09		1.80	35.72	0.04	0.05	0.09
1.80, 25.40	1.48	52.58	1.72	1.76	1.99	1.48	55.60	1.80, 25.40	0.12	0.24	0.39	1.80, 25.40	1.48	53.72	0.78	0.84	1.02
	1.54	50.20	1.16	1.17	1.32	1.54	52.94		0.00	0.10	0.21		1.54	51.16	0.46	0.50	0.62
	1.60	47.61	0.77	0.77	0.89	1.60	49.96		−0.03	0.04	0.12		1.60	48.34	0.26	0.29	0.38
	1.66	44.63	0.50	0.50	0.60	1.66	46.48		−0.04	0.04	0.10		1.66	45.10	0.15	0.17	0.24
	1.72	40.88	0.29	0.29	0.37	1.72	42.11		−0.02	0.03	0.09		1.72	41.08	0.06	0.08	0.14
1.72, 29.50	1.48	59.99	1.51	1.54	1.81	1.48	62.84	1.72, 29.50	0.02	0.15	0.31	1.72, 29.50	1.48	61.00	0.62	0.68	0.90
	1.54	56.48	0.95	0.96	1.15	1.54	58.84		0.00	0.09	0.21		1.54	57.22	0.33	0.38	0.52
	1.60	52.36	0.58	0.59	0.73	1.60	54.20		−0.05	0.02	0.11		1.60	52.82	0.16	0.19	0.30
	1.66	46.94	0.30	0.32	0.46	1.66	48.04		−0.03	0.03	0.12		1.66	47.08	0.07	0.09	0.20
1.64, 35.00	1.48	68.53	1.21	1.25	1.58	1.48	71.00	1.64, 35.00	−0.06	0.07	0.26	1.64, 35.00	1.48	69.30	0.43	0.51	0.76
	1.52	64.75	1.03	1.17	1.20	1.52	66.80		−0.08	0.03	0.18		1.52	65.30	0.26	0.32	0.52
	1.56	60.27	0.54	0.57	0.81	1.56	61.81		−0.07	0.02	0.15		1.56	60.58	0.14	0.19	0.37
	1.60	54.29	0.31	0.36	0.61	1.60	55.20		0.06	0.15	0.19		1.60	54.34	0.08	0.13	0.32
1.56, 44.00	1.48	79.07	0.83	0.93	1.39	1.48	80.87	1.56, 44.00	−0.10	0.05	0.31	1.56, 44.00	1.48	79.47	0.25	0.35	0.71
	1.50	75.25	0.63	0.72	1.29	1.50	76.69		−0.07	0.06	0.31		1.50	75.50	0.17	0.26	0.62
	1.52	70.63	0.50	0.62	1.06	1.52	71.70		−0.05	0.07	0.35		1.52	70.73	0.13	0.23	0.60
	1.54	64.38	0.32	0.56	1.10	1.54	65.03		−0.01	0.13	0.50		1.54	64.36	0.11	0.24	0.74

Flint Leading	n ₂	S_I			S_II

		F/3	F/4	F/5	F/3	F/4	F/5
n ₁ = 1.88	1.48	−0.76	−1.19	− 1.42	−12.71	−19.80	−22.60
	1.56	−0.33	−0.59	−0.74	−5.14	−9.43	−11.20
	1.64	−0.12	−0.31	− 0.40	−1.02	−4.09	−5.21
	1.72	0.00	−0.14	−0.21	1.47	−0.93	−1.79
	1.80	0.11	−0.02	−0.08	3.37	1.21	0.42
n ₁ = 1.80	1.48	−0.40	−0.81	−1.03	−4.18	−11.20	−13.80
	1.54	−0.16	−0.45	−0.60	−0.61	−5.51	−7.35
	1.60	−0.03	−0.24	−0.36	1.67	−2.07	− 3.44
	1.66	0.07	−0.11	−0.20	3.24	0.18	−0.94
	1.72	0.16	0.00	−0.09	4.64	1.85	0.83
n ₁ = 1.72	1.48	−0.08	−0.47	−0.68	3.17	−3.83	− 6.45
	1.54	0.07	−0.22	−0.36	4.67	−0.39	−2.27
	1.60	0.16	−0.06	−0.18	5.80	1.75	0.26
	1.66	0.27	0.07	−0.04	7.17	3.43	2.05
n ₁ = 1.64	1.48	0.23	−0.16	−0.36	9.57	2.32	−0.36
	1.52	0.29	−0.03	−0.20	9.54	3.40	1.20
	1.56	0.35	0.07	−0.08	9.88	4.56	2.61
	1.60	0.47	0.20	0.05	11.24	5.80	3.83
n ₁ = 1.56	1.48	0.57	0.16	−0.05	15.89	7.63	4.65
	1.50	0.60	0.23	0.02	15.80	7.98	5.15
	1.52	0.70	0.31	0.11	16.74	9.00	5.75
	1.54	0.89	0.47	0.25	20.37	10.50	9.04

Crown-Leading Glass Refractive Index	n ₁	S_I			S_II

		F/3	F/4	F/5	F/3	F/4	F/5
n ₂ = 1.88	1.48	0.14	0.01	0.00	−12.30	− 8.45	−7.95
	1.56	0.06	−0.02	−0.01	−7.17	−4.71	−4.59
	1.64	0.02	−0.02	−0.01	− 4.42	−2.70	−2.75
	1.72	0.02	−0.02	−0.01	−2.91	−1.59	−1.78
	1.80	0.03	−0.01	0.00	−2.30	−1.08	−1.11
n ₂ = 1.80	1.48	0.17	0.03	0.02	−12.30	−8.20	−7.62
	1.54	0.10	0.01	0.01	−8.17	−5.25	−4.98
	1.60	0.06	0.00	0.00	−5.68	−3.45	−3.34
	1.66	0.05	0.00	0.00	−4.17	−2.34	−2.29
	1.72	0.07	0.01	0.01	−3.77	−1.74	−1.65
n ₂ = 1.72	1.48	0.21	0.06	0.04	−12.30	−7.92	−7.20
	1.54	0.14	0.03	0.02	−8.29	−5.09	−4.68
	1.60	0.11	0.02	0.02	−6.02	−3.43	−3.14
	1.66	0.14	0.05	0.03	−5.10	−2.64	−2.30
n ₂ = 1.64	1.48	0.26	0.10	0.06	−12.40	−7.64	−6.73
	1.52	0.22	0.08	0.05	−9.77	−5.75	−5.03
	1.56	0.22	0.08	0.06	−8.16	−4.55	−3.89
	1.60	0.28	0.13	0.08	−7.86	−4.11	−3.30
n ₂ = 1.56	1.48	0.39	0.18	0.12	−13.40	−7.66	−6.33
	1.50	0.40	0.19	0.12	−12.40	−6.91	−5.59
	1.52	0.45	0.22	0.14	−12.20	−6.94	−5.12
	1.54	0.60	0.33	0.22	−13.80	−7.03	−5.19

Leading Type and m	Cemented-Doublet Glasses		R ₁ (mm)	R ₂ (mm)	R ₃ mm)	s (mm)	s′ (mm)
Crown leading ^b m = 0	PK1	SF2	60.93	−42.25	−123.23	Infinity	95.24
	BK7	SF12	61.79	−42.51	−134.11	Infinity	95.72
	BK8	SF5	61.96	−44.22	−130.40	Infinity	95.78
	SSK50	SF5	66.57	−38.20	−304.62	Infinity	95.14
	BK7	SF5	61.78	−44.18	−127.70	Infinity	95.80
	BaK5	SF8	63.86	−45.30	−160.51	Infinity	95.59
	K50	SF15	62.07	−45.80	−126.32	Infinity	95.85
	BaK4	SF15	64.46	−45.93	−170.71	Infinity	95.55
	K3	SF53	61.79	−47.06	−117.89	Infinity	95.98
	K10	SF14	60.79	−47.32	−103.59	Infinity	96.21
	KF3	SF11	61.48	−48.81	−108.08	Infinity	96.18
	BaF5	SF11	66.24	−52.49	−183.12	Infinity	95.58
Flint leading ^c m = 0	SF5	BaK50	46.03	24.39	1254.1	Infinity	93.39
	SF15	ZK1	42.12	24.97	3274.2	Infinity	93.10
	SF10	SSKN8	48.33	25.40	693.8	Infinity	93.37
	SF10	BaF9	50.69	24.87	580.9	Infinity	93.48
	SF5	BaK5	45.01	24.50	1545.0	Infinity	93.34
	SF2	BK7	42.01	23.93	4752.8	Infinity	93.20
Flint leading ^d m = −1	SF16	BK7	64.25	30.81	−121.87	−198.30	195.82
	SF2	BaK1	77.06	29.51	−134.22	−198.05	196.22
	SF2	SSK3	91.09	25.40	−142.46	−197.84	196.54
	SF2	BK7a	64.14	30.87	−121.87	−198.31	195.81
	SF15	BaF9	90.35	29.27	−149.95	−198.01	196.49
	SF15	SSK3	81.88	31.29	−144.01	−198.13	196.30
	SF10	BaF9	85.75	32.07	−150.56	−198.15	196.38
	SFL6	BaLF8	62.27	35.06	−129.35	−198.81	195.57

Abstract

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Figures (9)

Tables (9)

Equations (19)

Applied Optics