Basic theory of a four-lens optical system design

Josif Sakin; Lev Sakin

doi:10.1364/AO.35.000572

Applied Optics
Vol. 35,
Issue 4,
pp. 572-581
(1996)
•https://doi.org/10.1364/AO.35.000572

Basic theory of a four-lens optical system design

Josif Sakin and Lev Sakin

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Josif Sakin and Lev Sakin, "Basic theory of a four-lens optical system design," Appl. Opt. 35, 572-581 (1996)

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Abstract

The theory and the method of calculation of an optical system based on aberration parameters C, W, P are presented. These parameters define all the aberrations of optical systems. The method is based on the synthesis of multilens systems from the single elements. The formulas for the relationships of increments between the angles of the first paraxial ray and the increments in the basic parameters are introduced. These formulas permit the parameters to vary and correct the aberrations. These angle calculations of the first paraxial ray of a four-lens cemented and uncemented system are presented. It is a method for designing complex optical systems from the initial stage to the final stage with corrected aberrations.

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Equations (71)

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Coefficients	Formulas	Values	Assigned Coefficient

1	2	3	4
C			0.0
C ₁			0.01
C ₂			−0.02730
α	$\frac{C - C_{2}}{C_{1} - C_{2}}$	0.73190
n ₄			1.51680
n ₅			1.61659
K ₁			−0.19490
K ₂			0.10058
K ₃			−0.23565
K ₄			0.61782
K ₅	$- \frac{α n_{4} K_{1} (n_{5} - 1)}{(1 - α) n_{5} K_{3} (1 - n_{4})}$	2.52756
K ₆	$\frac{α (1 - n_{4} K_{2}) (n_{5} - 1)}{(1 - α) n_{5} K_{3} (1 - n_{4})} - \frac{K_{4}}{K_{3}}$	9.86722
K ₇	α² + (1 − α)² K ₅	0.71735
K ₈	${(1 - α)}^{2} K_{6} + α (1 - α) (2 + \frac{1}{n_{5}})$	1.22305
K ₉	$\frac{1}{K_{7}}$	1.39401
K ₁₀	$- \frac{K_{8}}{K_{7}}$	−1.70495
K ₁₁	K ₅ K ₉	3.52346
K ₁₂	K ₅ K ₁₀ + K ₆	5.55786
K ₁₃			0.81685
K ₁₄			0.22524
K ₁₅			−19.3901
K ₁₆			0.85391
K ₁₇			−0.23612
K ₁₈			1.62306
K ₁₉	α³ K ₉ ² K ₁₃ + (1 − α)³ K ₁₁ ² K ₁₆	0.82663
K ₂₀	α³(2K ₉ K ₁₀ K ₁₃ + K ₉ K ₁₄) + (1 − α)³2K ₁₁ K ₁₂ K ₁₆ + K ₁₁ K ₁₇) + 4α(1 − α)2 K ₁₁	−0.02932
K ₂₁	$\begin{array}{l} α^{3} ({K_{10}}^{2} K_{13} + K_{10} K_{14} + K_{15}) \\ + {(1 - α)}^{3} ({K_{12}}^{2} K_{16} \\ + K_{12} K_{17} + K_{18}) \\ + 4 α {(1 - α)}^{2} K_{12} + α (1 - α) \\ \times [2 α (2 + \frac{1}{n_{5}}) - 1] \end{array}$	−4.58203
K ₂₂			−0.194490
K ₂₃			0.33356
K ₂₄			−0.21818
K ₂₅			−0.08350
K ₂₆	K ₉ K ₂₂α	−0.19885
K ₂₇	(K ₁₀ K ₂₂ + K ₂₃)α	0.48734
K ₂₈	K ₉ K ₂₄α	−0.22260
K ₂₉	(K ₁₀ K ₂₄ + K ₂₅)α	0.21114
K ₃₀	K ₁ K ₉α	−0.19885
K ₃₁	(K ₁ K ₁₀ + K ₂)α	0.31682
K ₃₂	K ₃ K ₁₁(1 − α)	−0.22260
K ₃₃	$(K_{3} K_{12} + K_{4}) (1 - α) + \frac{α}{n_{5}}$	0.26725

Coefficients	Formulas	Values	Assigned Values

1	2	3	4
K ₃₄	2K ₁₃ K ₉	2.27739
K ₃₅	2K ₁₃ K ₁₀ + K ₁₄	−2.56014
K ₃₆	(1 − α)2K ₁₆ K ₁₁	1.61327
K ₃₇	(1 − α)(2K ₁₆ K ₁₂ + K ₁₇) + 4α	5.40906
K ₃₈			−18.625
K ₃₉			9.262
K ₄₀			3.126
K ₄₁	α² K ₃₄ K ₃₈	−22.72151
K ₄₂	α² K ₃₄ K ₃₉	11.29915
K ₄₃	α² K ₃₄ K ₄₀	3.81355
K ₄₄	α² K ₃₅ K ₃₈	25.54250
K ₄₅	α² K ₃₅ K ₃₉	−12.70200
K ₄₆	α² K ₃₅ K ₄₀	−4.28703
K ₄₇	αK ₁₃ K ₃₈ ²	207.38944
K ₄₈	2αK ₁₃ K ₃₈ K ₃₉	−206.26480
K ₄₉	αK ₁₃ K ₃₉ ²	51.28656
K ₅₀	2αK ₁₃ K ₃₈ K ₄₀	−69.61604
K ₅₁	2αK ₁₃ K ₃₉ K ₄₀	34.61926
K ₅₂	αK ₁₃ K ₄₀ ²	5.84214
K ₅₃	$\frac{(1 - α) K_{36}}{K_{3}}$	−1.83594
K ₅₄	$\frac{(1 - α) K_{37}}{K_{3}}$	−6.15391
K ₅₅	$\frac{(1 - α) K_{16}}{{K_{3}}^{2}}$	4.12263

Glass	$\frac{1}{n}$	$Δ \frac{1}{n}$	$\frac{Δ n_{c}}{n}$	$Δ \frac{Δ n_{c}}{n}$	α	Δα	$\frac{Δ α}{Δ \frac{1}{n}}$	${(\frac{Δ α}{Δ \frac{1}{n}})}^{2}$	$\frac{α}{n}$	$Δ \frac{α}{n}$	W	P	C

1	2	3	4	5	6	7	8	9	10	11	12	13	14
	1		0		0			0
		−0.34072		0.00531		0.64938	−1.9059	3.63247		0.42812	−0.81595	1.55514	−0.01012
BK7	0.65928		0.00531		0.64938				0.42812
		0.34072		−0.00531		2.14972	6.30934	39.80783		2.37098	14.9593	94.3834	−0.03349
	1		0		2.79910				2.79910
		−0.38141		0.01041		−2.39298	6.27404	39.36352		−2.54788	−15.9855	−100.294	0.06533
F4	0.61859		0.01041		0.40612				0.25122
		0.38141		−0.01041		0.79555	2.08581	4.35062		0.94045	1.98245	4.13504	−0.02172
	1		0		1.20167				1.20167
		−0.34072		0.00531		−0.4360	1.27964	1.63748		−0.69688	−0.89175	−1.14112	0.00679
BK7	0.65928		0.00531		0.76567				0.50479
		0.34072		−0.00531		−0.03377	−0.09911	0.00982		0.22711	−0.00225	0.00223	0.00053
	1		0		0.73190				0.73190
		−0.38141		0.01041		−0.31690	0.83086	0.69034		−0.47519	−0.39482	−0.32804	0.00865
F4	0.61859		0.01041		0.4150				0.25671
		0.38141		−0.01041		0.5850	1.53378	2.53249		0.73429	1.14004	1.74858	−0.001597
	1		0		1				1		W	P	C
											−0.00846	0.0617	0.000

Glass	n	Δn	α	αn	Δ(αn)	$\frac{Δ n}{Δ (α n)}$	d	αd	h − αd	r

1	2	3	4	5	6	7	8	9	10	11
	1		0	0
		0.51680			0.98498	0.52468			100	52.468
BK7	1.51680		0.64938	0.98498			5	3.247
		−0.51680			1.81412	−0.28488			96.753	−27.563
	1		2.79910	2.79910			0.5	1.399
		0.61659			−2.14257	−0.28778			95.354	− 27.441
F4	1.61659		0.40612	0.65653			2	0.812
		−0.61659			0.54514	− 1.13107			94.542	−106.933
	1		1.20167	1.20167			0.5	0.601
		0.51680			−0.04030	− 12.8238			93.941	−1204.68
BK7	1.51680		0.76567	1.16137			2	1.531
		−0.51680			0.57053	− 0.90582			92.410	−83.706
	1		0.73190	1.73190			0.5	0.362
		0.61659			− 1.06101	−0.58113			92.048	−53.492
F4	1.61659		0.41500	0.67088			2	0.658
		−0.61659			0.32912	−1.87345			91.390	− 171.214
	1	1	1

Radius (mm)	Thickness (mm)	Glass	Diameter (mm)
138.49044	3.50	BK7	127.6
93.54112	21.081929	Air	120.6
161.05059	220.00	BK7	118.0
−149.28981	0.10	Air	27.4
Stop ∞	0.10	Air	26.8
68.92527	103.00	BK7	27.6
2241.21673	7.042146	Air	46.6
−42.90263	2.00	BK7	46.6
4945.08373	1.292768	Air	52.2

Abstract

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Tables (5)

Equations (71)

Applied Optics