Aspherical surfaces used to minimize oblique astigmatic error, power error, and distortion of some high positive and negative power ophthalmic lenses

Milton Katz

doi:10.1364/AO.21.002982

Applied Optics
Vol. 21,
Issue 16,
pp. 2982-2991
(1982)
•https://doi.org/10.1364/AO.21.002982

Aspherical surfaces used to minimize oblique astigmatic error, power error, and distortion of some high positive and negative power ophthalmic lenses

Milton Katz

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Milton Katz, "Aspherical surfaces used to minimize oblique astigmatic error, power error, and distortion of some high positive and negative power ophthalmic lenses," Appl. Opt. 21, 2982-2991 (1982)

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Abstract

Ophthalmic lenses are generally corrected for oblique astigmatic error or optimized to balance astigmatic and power errors. Distortion is not corrected in these lenses. The nature of the aspheric surface needed to minimize astigmatic error, power error, and distortion in −20-, −14-, and +14-diopter lenses was investigated using the accos V lens design programs. The best correction for the −20- and −14-diopter lenses was obtained by allowing their front and rear surfaces to become deformed ellipsoids of revolution about the minor axis. The +14-diopter front and rear lens surfaces became deformed ellipsoids of revolution about the major and minor axes, respectively. Performance data over the full field of view, corresponding to ±30° of eye rotation, are given.

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Corrections

Milton Katz, "Aspherical surfaces used to minimize oblique astigmatic error, power error, and distortion of some high positive and negative power ophthalmic lenses: erratum," Appl. Opt. 21, 4399_1-4399 (1982)
https://opg.optica.org/ao/abstract.cfm?uri=ao-21-24-4399_1

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Data for −20-Diopter Lenses

	Lens forms

Lens data	Ostwalt	Wollaston	Minimum distortion	Aspheric A	Aspheric B
First surface power F₁ (diopter)	−0.00010	8.621	14.418	14.418	14.418
First surface radius r₁ (mm)	−5.23 × 10⁶	60.66847	36.27501	36.27501	36.27501
Second surface power F₂ (diopter)	−19.99990	28.650	−35.500	−35.500	−35.500
Second surface radius r₂ (mm)	26.15013	18.2548	15.15942	15.15942	15.15942
Thickness (mm)	0.600	0.600	0.600	0.600	0.600
Lens diameter (mm)	32.000	32.000	32.000	32.000	32.000
Object field angle θ (deg)	50.251	46.372	45.754	41.641a	41.641a
Image field angle θ′ (deg)	30.000	30.000	30.000	30.003	30.024
Astigmatic error (OAE) at 30° (diopter)	−0.006	−0.008	−4.317	0.028	−0.051
Power error (MOE) at 30° (diopter)	−0.052	−0.046	−4.000	−0.397	−0.687
Distortion at 30° (%)	−26.059	−15.36	−13.899	−0.554	−0.473

a Required value of θ_e for correcting distortion.

Oblique Astigmatism at Fractional Field Angles of −20-Dlopter Lenses

	Oblique astigmatic error (diopter)

Fractional Field angle	Ostwalt form	Wollaston form	Minimum distortion	Aspheric A	Aspheric B
1.0	−0.006	−0.008	−4.317	0.028	−0.051
0.9	−0.005	−0.043	−3.639	−0.469	0.588
0.8	−0.005	−0.068	−2.987	0.099	0.512
0.7	−0.004	−0.081	−2.370	0.428	0.148
0.6	−0.003	−0.083	−1.800	0.276	−0.043
0.5	−0.003	−0.074	−1.287	0.060	0.034
0.4	−0.002	−0.058	−0.845	0.110	0.141
0.3	−0.001	−0.040a	−0.484	0.257	0.130
0.2	−0.001	−0.022a	−0.219	0.246	0.060a
0.1	−0.000	−0.006	−0.055	0.091	0.014

a Interpolated values.

Power Error at Fractional Field Angles of −20-Diopter Lenses

	Mean oblique error (diopter)

Fractional field angle	Ostwalt form	Wollaston form	Minimum distortion	Aspheric A	Aspheric B
1.0	−0.052	−0.046	−3.999	−0.397	−0.687
0.9	−0.047	−0.077	−3.406	−0.414	−0.165
0.8	−0.042	−0.097	−2.833	0.076	−0.034
0.7	−0.036	−0.106	−2.277	0.365	−0.068
0.6	−0.029	−0.103	−1.751	0.366	−0.034
0.5	−0.022	−0.090	−1.267	0.303	0.078
0.4	−0.016	−0.069	−0.842	0.348	0.149
0.3	−0.010	−0.049a	−0.484	0.384	0.121
0.2	−0.005	−0.229a	+0.222	0.280	0.015a
0.1	−0.001	−0.006	−0.056	0.093	0.014

a Interpolated values.

Distortion at Fractional Field Angles of −20-Diopter Lenses

	Distortion (%)

Fractional field angle	Ostwalt form	Wollaston form	Minimum distortion	Aspheric A	Aspheric B
1.0	−26.06	−15.54	−13.89	−0.55	−0.47
0.9	−22.66	−13.31	−11.88	−0.32	−0.36
0.8	−19.17	−11.11	−9.89	−0.30	−0.53
0.7	−15.68	−896	−7.96	−0.48	−0.79
0.6	−12.25	−6.91	−6.13	−0.69	−0.96
0.5	−8.99	−5.01	−4.44	−0.69	−0.97
0.4	−6.04	−3.33	−2.92	−0.49	−0.82
0.3	−3.54	−2.00	−1.71	−0.24	−0.30
0.2	−1.62	−1.00	−0.78	−0.08	−0.15
0.1	−0.42	−0.22	−0.20	−0.01	−0.08

Summary of Aberrations vs Field angles of −20-Diopter Aspheric Lenses

	Aspheric A			Aspheric B

Fractional field angle	OAE (diopter)	MOE (diopter)	Dist. (%)	OAE (diopter)	MOE (diopter)	Dist. (%)
1.0	0.028	−0.397	−0.554	−0.051	−0.687	−0.473
0.9	−0.469	−0.414	−0.322	0.588	−0.165	−0.364
0.8	−0.099	−0.076	−0.301	0.512	−0.034	−0.530
0.7	0.428	0.365	−0.475	0.148	−0.068	−0.786
0.6	0.276	0.366	−0.688	−0.043	−0.034	−0.957
0.5	0.060	0.303	−0.689	0.034	0.078	−0.965
0.4	0.110	0.348	−0.490	0.141	0.149	−0.820
0.3	0.257	0.384	−0.243	0.130	0.121	−0.299
0.2	0.246	0.280	−0.076	0.060a	0.059a	−0.151a
0.1	0.091	0.093	−0.011	0.015	0.014	−0.082

a Interpolated values.

Surface Profiles of −20-Diopter Aspherics

	Aspheric A		Aspheric B

	Surface 1	Surface 2	Surface 1	Surface 2
Radius	36.27501 mm	15.15942 mm	36.27501 mm	15.15942 mm
cc	3.68865E + 00	2.52690E − 01	3.05048E + 00	1.76047E − 02
d	5.33797E − 05	4.42675E − 05	6.05353E − 05	6.71999E − 05
e	5.43857E − 08	4.00319E − 07	5.35469E − 08	3.20996E − 07
f	−9.28519E − 11	3.99395E − 10	−7.13575E − 11	1.52903E − 09
g	−1.36799E −12	5.81397E − 12	1.20656E − 12	8.86909E − 12

Data for −14-Diopter Lenses

	Lens forms

Lens data	Ostwalt	Wollaston	Minimum distortion	Aspheric
First surface power F₁ (diopter)	0.69981	12.78528	16.24536	16.24536
First surface radius r₁ (mm)	747.34890	40.90643	32.19381	32.19381
Second surface power F₂ (diopter)	−14.70000	−26.85001	−30.35000	−30.35000
Second surface radius r₂ (mm)	35.57823	19.47858	17.23229	17.23229
Thickness (mm)	0.600	0.600	0.600	0.600
Lens diameter (mm)	40.000	38.000	36.000	32.000
Object field angle θ (deg)	43.85248	40.82617	40.66692	38.32648a
Image field angle θ′ (deg)	30.000	30.000	30.000	30.011607
Astigmatic error (OAE) at 30° (diopter)	−0.0005	0.0058	−1.4586	0.02645
Power error (MOE) at 30° (diopter)	0.242	0.245	−1.1505	−0.05682
Distortion at 30°	−17.212	−8.379	−7.989	0.046

a Required value of θ_e for correcting distortion.

Oblique Astigmatism at Fractional Field Angles for −14-Diopter Lenses

	Oblique astigmatic error (diopter)

Fractional field angles	Ostwalt	Wollaston	Minimum distortion	Aspheric form
1.0	−0.0005	0.006	−1.459	0.026
0.9	−0.010	−0.003	−1.238	−0.105
0.8	−0.017	−0.010	−1.022	0.035
0.7	−0.021	−0.013	−0.816	0.111
0.6	−0.022	−0.014	−0.622	0.121
0.5	−0.020	−0.013	−0.447	−0.004
0.4	−0.016	−0.011	−0.296	−0.067
0.3	−0.011a	−0.007	−0.169	0.004
0.2	−0.005	−0.003	−0.077	0.054
0.1	−0.001a	−0.001	−0.019	0.029

a Interpolated values.

Power Error at Fractional Field Angles for −14-Diopter Lenses

	Mean oblique error (diopter)

Fractional field angles	Ostwalt form	Wollaston form	Minimum distortion	Aspheric form
1.0	0.2423	0.2541	−1.1505	−0.0568
0.9	0.2051	0.2092	−0.9826	−0.1116
0.8	0.1688	0.1674	−0.8168	0.0357
0.7	0.1364	0.1291	−0.6559	0.1364
0.6	0.1016	0.0951	−0.5036	0.1466
0.5	0.0724	0.0660	−0.3638	0.0800
0.4	0.0473	0.0423	−0.2419	0.0515
0.3	0.0297a	0.0235	−0.1392	0.0681a
0.2	0.0121	0.0104	−0.6320	0.0847
0.1	0.0030a	0.0026	−0.0160	0.0335

a Interpolated values.

Distortion at Fractional Field Angles for −14-Diopter Lenses

	Distortion (%)

Fractional field angles	Ostwalt form	Wollaston form	Minimum distortion	Aspheric form
1.0	−17.2116	−8.3790	−7.9889	0.0460
0.9	−14.6321	−7.0580	−6.7248	0.1479
0.8	−12.1028	−5.7886	−5.5121	0.1220
0.7	−9.8380	−4.5897	−4.3678	−0.0251
0.6	−7.3850	−3.4819	−3.3119	−0.2236
0.5	−5.3074	−2.4881	−2.3655	−0.3506
0.4	−3.4972	−1.6319	−1.5508	−0.3301
0.3	−2.0134	−0.9357	−0.8891	−0.2010a
0.2	−0.9105	−0.4222	−0.4011	−0.0845
0.1	−0.2300	−0.1064	−0.1012	−0.0178

a Interpolated values.

Profiles of −14-Diopter Aspheric Lens Surfaces

	First surface	Second surface
Radius	32.19381 mm	17.23229 mm
cc	2.16933E + 00	6.66319E − 02
d	7.90881E − 05	8.63237E − 05
e	−2.15276E − 08	3.08554E − 07
f	−1.98961E − 10	−2.24752E − 10
g	−4.39336E − 13	3.46721E − 12

Data for +14-Diopter Lenses

	Lens forms

Lens data	Minimum astigmatism	Minimum distortion	Aspheric
First surface power F₁ (diopter)	17.12425	19.41926	17.12425
First surface radius r₁ (mm)	30.53149	26.93202	30.54149
Second surface power F₂ (diopter)	−5.10000	−8.00000	−5.10000
Second surface radius r₂ (mm)	102.54902	65.37500	102.54902
Thickness (mm)	9.200	9.200	9.200
Lens diameter (mm)	40.000	40.000	40.000
Object field angle θ (deg)	16.17261	16.03651	17.84671a
Image field angle θ′ (deg)	30.000	30.000	30.231
Astigmatic error (OAE) at 30° (diopter)	1.927	2.279	−0.304
Power error (MOE) at 30° (diopter)	1.056	1.397	−1.781
Distortion at 30° (%)	11.019	10.281	0.914

a Required value of θ_e for correcting distortion.

Oblique Astigmatism at Fractional Field Angles of +14-Diopter Lenses

	Oblique astigmatic errors (diopter)

Fractional field angles	Minimum astigmatism	Minimum distortion	Aspheric form
1.0	1.927	2.279	−0.304
0.9	1.500	1.792	−0.627
0.8	1.145	1.381	−0.749
0.7	0.853	1.034	−0.783
0.6	0.612	0.747	−0.759
0.5	0.417	0.511	−0.671
0.4	0.263	0.323	−0.530
0.3	0.146	0.180	−0.351
0.2	0.064	0.080	−0.175
0.1	0.016	0.020	−0.047

Mean Oblique Error at Fractional Field Angles of +14-Diopter Lenses

	Mean oblique error (diopter)

Fractional field angle	Minimum astigmatism	Minimum distortion	Aspheric form
1.0	1.056	1.397	−1.781
0.9	0.814	1.093	−1.776
0.8	0.617	0.839	−1.649
0.7	0.456	0.626	−1.466
0.6	0.325	0.450	−1.247
0.5	0.221	0.307	−0.995
0.4	0.139	0.194	−0.721
0.3	0.077	0.108	−0.449
0.2	0.034	0.047	−0.215
0.1	−0.028	0.012	−0.056

Distortion at Fractional Field Angles of +14-Diopter Lenses

	Distortion (%)

Fractional field angle	Minimum astigmatism	Minimum distortion	Aspheric form
1.0	11.019	10.285	0.914
0.9	8.601	8.061	0.120
0.8	6.586	6.193	−0.397
0.7	4.911	4.631	−0.706
0.6	3.530	3.342	−0.834
0.5	2.407	2.281	−0.806
0.4	1.519	1.442	−0.657
0.3	0.845	0.805	−0.440
0.2	0.373	0.357	−0.221
0.1	0.093	0.092	−0.059

Surface Profile of +14-Diopter Aspheric Lens

	First surface	Second surface
Radius	30.54149 mm	102.54902 mm
cc	−3.66537E − 01	−1.56764E + 01
d	−3.64926E − 06	−1.84671E − 06
e	−2.91705E − 09	−7.99216E − 09
f	−5.56478E − 12	−1.46071E − 11
g	−3.34229E − 14	6.64155E − 15

Table I

Data for −20-Diopter Lenses

	Lens forms

Lens data	Ostwalt	Wollaston	Minimum distortion	Aspheric A	Aspheric B
First surface power F₁ (diopter)	−0.00010	8.621	14.418	14.418	14.418
First surface radius r₁ (mm)	−5.23 × 10⁶	60.66847	36.27501	36.27501	36.27501
Second surface power F₂ (diopter)	−19.99990	28.650	−35.500	−35.500	−35.500
Second surface radius r₂ (mm)	26.15013	18.2548	15.15942	15.15942	15.15942
Thickness (mm)	0.600	0.600	0.600	0.600	0.600
Lens diameter (mm)	32.000	32.000	32.000	32.000	32.000
Object field angle θ (deg)	50.251	46.372	45.754	41.641a	41.641a
Image field angle θ′ (deg)	30.000	30.000	30.000	30.003	30.024
Astigmatic error (OAE) at 30° (diopter)	−0.006	−0.008	−4.317	0.028	−0.051
Power error (MOE) at 30° (diopter)	−0.052	−0.046	−4.000	−0.397	−0.687
Distortion at 30° (%)	−26.059	−15.36	−13.899	−0.554	−0.473

a Required value of θ_e for correcting distortion.

Table II

Oblique Astigmatism at Fractional Field Angles of −20-Dlopter Lenses

	Oblique astigmatic error (diopter)

Fractional Field angle	Ostwalt form	Wollaston form	Minimum distortion	Aspheric A	Aspheric B
1.0	−0.006	−0.008	−4.317	0.028	−0.051
0.9	−0.005	−0.043	−3.639	−0.469	0.588
0.8	−0.005	−0.068	−2.987	0.099	0.512
0.7	−0.004	−0.081	−2.370	0.428	0.148
0.6	−0.003	−0.083	−1.800	0.276	−0.043
0.5	−0.003	−0.074	−1.287	0.060	0.034
0.4	−0.002	−0.058	−0.845	0.110	0.141
0.3	−0.001	−0.040a	−0.484	0.257	0.130
0.2	−0.001	−0.022a	−0.219	0.246	0.060a
0.1	−0.000	−0.006	−0.055	0.091	0.014

a Interpolated values.

Table III

Power Error at Fractional Field Angles of −20-Diopter Lenses

	Mean oblique error (diopter)

Fractional field angle	Ostwalt form	Wollaston form	Minimum distortion	Aspheric A	Aspheric B
1.0	−0.052	−0.046	−3.999	−0.397	−0.687
0.9	−0.047	−0.077	−3.406	−0.414	−0.165
0.8	−0.042	−0.097	−2.833	0.076	−0.034
0.7	−0.036	−0.106	−2.277	0.365	−0.068
0.6	−0.029	−0.103	−1.751	0.366	−0.034
0.5	−0.022	−0.090	−1.267	0.303	0.078
0.4	−0.016	−0.069	−0.842	0.348	0.149
0.3	−0.010	−0.049a	−0.484	0.384	0.121
0.2	−0.005	−0.229a	+0.222	0.280	0.015a
0.1	−0.001	−0.006	−0.056	0.093	0.014

a Interpolated values.

Table IV

Distortion at Fractional Field Angles of −20-Diopter Lenses

	Distortion (%)

Fractional field angle	Ostwalt form	Wollaston form	Minimum distortion	Aspheric A	Aspheric B
1.0	−26.06	−15.54	−13.89	−0.55	−0.47
0.9	−22.66	−13.31	−11.88	−0.32	−0.36
0.8	−19.17	−11.11	−9.89	−0.30	−0.53
0.7	−15.68	−896	−7.96	−0.48	−0.79
0.6	−12.25	−6.91	−6.13	−0.69	−0.96
0.5	−8.99	−5.01	−4.44	−0.69	−0.97
0.4	−6.04	−3.33	−2.92	−0.49	−0.82
0.3	−3.54	−2.00	−1.71	−0.24	−0.30
0.2	−1.62	−1.00	−0.78	−0.08	−0.15
0.1	−0.42	−0.22	−0.20	−0.01	−0.08

Table V

Summary of Aberrations vs Field angles of −20-Diopter Aspheric Lenses

	Aspheric A			Aspheric B

Fractional field angle	OAE (diopter)	MOE (diopter)	Dist. (%)	OAE (diopter)	MOE (diopter)	Dist. (%)
1.0	0.028	−0.397	−0.554	−0.051	−0.687	−0.473
0.9	−0.469	−0.414	−0.322	0.588	−0.165	−0.364
0.8	−0.099	−0.076	−0.301	0.512	−0.034	−0.530
0.7	0.428	0.365	−0.475	0.148	−0.068	−0.786
0.6	0.276	0.366	−0.688	−0.043	−0.034	−0.957
0.5	0.060	0.303	−0.689	0.034	0.078	−0.965
0.4	0.110	0.348	−0.490	0.141	0.149	−0.820
0.3	0.257	0.384	−0.243	0.130	0.121	−0.299
0.2	0.246	0.280	−0.076	0.060a	0.059a	−0.151a
0.1	0.091	0.093	−0.011	0.015	0.014	−0.082

a Interpolated values.

Table VI

Surface Profiles of −20-Diopter Aspherics

	Aspheric A		Aspheric B

	Surface 1	Surface 2	Surface 1	Surface 2
Radius	36.27501 mm	15.15942 mm	36.27501 mm	15.15942 mm
cc	3.68865E + 00	2.52690E − 01	3.05048E + 00	1.76047E − 02
d	5.33797E − 05	4.42675E − 05	6.05353E − 05	6.71999E − 05
e	5.43857E − 08	4.00319E − 07	5.35469E − 08	3.20996E − 07
f	−9.28519E − 11	3.99395E − 10	−7.13575E − 11	1.52903E − 09
g	−1.36799E −12	5.81397E − 12	1.20656E − 12	8.86909E − 12

Table VII

Data for −14-Diopter Lenses

	Lens forms

Lens data	Ostwalt	Wollaston	Minimum distortion	Aspheric
First surface power F₁ (diopter)	0.69981	12.78528	16.24536	16.24536
First surface radius r₁ (mm)	747.34890	40.90643	32.19381	32.19381
Second surface power F₂ (diopter)	−14.70000	−26.85001	−30.35000	−30.35000
Second surface radius r₂ (mm)	35.57823	19.47858	17.23229	17.23229
Thickness (mm)	0.600	0.600	0.600	0.600
Lens diameter (mm)	40.000	38.000	36.000	32.000
Object field angle θ (deg)	43.85248	40.82617	40.66692	38.32648a
Image field angle θ′ (deg)	30.000	30.000	30.000	30.011607
Astigmatic error (OAE) at 30° (diopter)	−0.0005	0.0058	−1.4586	0.02645
Power error (MOE) at 30° (diopter)	0.242	0.245	−1.1505	−0.05682
Distortion at 30°	−17.212	−8.379	−7.989	0.046

a Required value of θ_e for correcting distortion.

Table VIII

Oblique Astigmatism at Fractional Field Angles for −14-Diopter Lenses

	Oblique astigmatic error (diopter)

Fractional field angles	Ostwalt	Wollaston	Minimum distortion	Aspheric form
1.0	−0.0005	0.006	−1.459	0.026
0.9	−0.010	−0.003	−1.238	−0.105
0.8	−0.017	−0.010	−1.022	0.035
0.7	−0.021	−0.013	−0.816	0.111
0.6	−0.022	−0.014	−0.622	0.121
0.5	−0.020	−0.013	−0.447	−0.004
0.4	−0.016	−0.011	−0.296	−0.067
0.3	−0.011a	−0.007	−0.169	0.004
0.2	−0.005	−0.003	−0.077	0.054
0.1	−0.001a	−0.001	−0.019	0.029

a Interpolated values.

Table IX

Power Error at Fractional Field Angles for −14-Diopter Lenses

	Mean oblique error (diopter)

Fractional field angles	Ostwalt form	Wollaston form	Minimum distortion	Aspheric form
1.0	0.2423	0.2541	−1.1505	−0.0568
0.9	0.2051	0.2092	−0.9826	−0.1116
0.8	0.1688	0.1674	−0.8168	0.0357
0.7	0.1364	0.1291	−0.6559	0.1364
0.6	0.1016	0.0951	−0.5036	0.1466
0.5	0.0724	0.0660	−0.3638	0.0800
0.4	0.0473	0.0423	−0.2419	0.0515
0.3	0.0297a	0.0235	−0.1392	0.0681a
0.2	0.0121	0.0104	−0.6320	0.0847
0.1	0.0030a	0.0026	−0.0160	0.0335

a Interpolated values.

Table X

Distortion at Fractional Field Angles for −14-Diopter Lenses

	Distortion (%)

Fractional field angles	Ostwalt form	Wollaston form	Minimum distortion	Aspheric form
1.0	−17.2116	−8.3790	−7.9889	0.0460
0.9	−14.6321	−7.0580	−6.7248	0.1479
0.8	−12.1028	−5.7886	−5.5121	0.1220
0.7	−9.8380	−4.5897	−4.3678	−0.0251
0.6	−7.3850	−3.4819	−3.3119	−0.2236
0.5	−5.3074	−2.4881	−2.3655	−0.3506
0.4	−3.4972	−1.6319	−1.5508	−0.3301
0.3	−2.0134	−0.9357	−0.8891	−0.2010a
0.2	−0.9105	−0.4222	−0.4011	−0.0845
0.1	−0.2300	−0.1064	−0.1012	−0.0178

a Interpolated values.

Table XI

Profiles of −14-Diopter Aspheric Lens Surfaces

	First surface	Second surface
Radius	32.19381 mm	17.23229 mm
cc	2.16933E + 00	6.66319E − 02
d	7.90881E − 05	8.63237E − 05
e	−2.15276E − 08	3.08554E − 07
f	−1.98961E − 10	−2.24752E − 10
g	−4.39336E − 13	3.46721E − 12

Table XII

Data for +14-Diopter Lenses

	Lens forms

Lens data	Minimum astigmatism	Minimum distortion	Aspheric
First surface power F₁ (diopter)	17.12425	19.41926	17.12425
First surface radius r₁ (mm)	30.53149	26.93202	30.54149
Second surface power F₂ (diopter)	−5.10000	−8.00000	−5.10000
Second surface radius r₂ (mm)	102.54902	65.37500	102.54902
Thickness (mm)	9.200	9.200	9.200
Lens diameter (mm)	40.000	40.000	40.000
Object field angle θ (deg)	16.17261	16.03651	17.84671a
Image field angle θ′ (deg)	30.000	30.000	30.231
Astigmatic error (OAE) at 30° (diopter)	1.927	2.279	−0.304
Power error (MOE) at 30° (diopter)	1.056	1.397	−1.781
Distortion at 30° (%)	11.019	10.281	0.914

a Required value of θ_e for correcting distortion.

Table XIII

Oblique Astigmatism at Fractional Field Angles of +14-Diopter Lenses

	Oblique astigmatic errors (diopter)

Fractional field angles	Minimum astigmatism	Minimum distortion	Aspheric form
1.0	1.927	2.279	−0.304
0.9	1.500	1.792	−0.627
0.8	1.145	1.381	−0.749
0.7	0.853	1.034	−0.783
0.6	0.612	0.747	−0.759
0.5	0.417	0.511	−0.671
0.4	0.263	0.323	−0.530
0.3	0.146	0.180	−0.351
0.2	0.064	0.080	−0.175
0.1	0.016	0.020	−0.047

Table XIV

Mean Oblique Error at Fractional Field Angles of +14-Diopter Lenses

	Mean oblique error (diopter)

Fractional field angle	Minimum astigmatism	Minimum distortion	Aspheric form
1.0	1.056	1.397	−1.781
0.9	0.814	1.093	−1.776
0.8	0.617	0.839	−1.649
0.7	0.456	0.626	−1.466
0.6	0.325	0.450	−1.247
0.5	0.221	0.307	−0.995
0.4	0.139	0.194	−0.721
0.3	0.077	0.108	−0.449
0.2	0.034	0.047	−0.215
0.1	−0.028	0.012	−0.056

Table XV

Distortion at Fractional Field Angles of +14-Diopter Lenses

	Distortion (%)

Fractional field angle	Minimum astigmatism	Minimum distortion	Aspheric form
1.0	11.019	10.285	0.914
0.9	8.601	8.061	0.120
0.8	6.586	6.193	−0.397
0.7	4.911	4.631	−0.706
0.6	3.530	3.342	−0.834
0.5	2.407	2.281	−0.806
0.4	1.519	1.442	−0.657
0.3	0.845	0.805	−0.440
0.2	0.373	0.357	−0.221
0.1	0.093	0.092	−0.059

Table XVI

Surface Profile of +14-Diopter Aspheric Lens

	First surface	Second surface
Radius	30.54149 mm	102.54902 mm
cc	−3.66537E − 01	−1.56764E + 01
d	−3.64926E − 06	−1.84671E − 06
e	−2.91705E − 09	−7.99216E − 09
f	−5.56478E − 12	−1.46071E − 11
g	−3.34229E − 14	6.64155E − 15

Aspherical surfaces used to minimize oblique astigmatic error, power error, and distortion of some high positive and negative power ophthalmic lenses

Author Affiliations

Abstract

Corrections

Cited By

Figures (5)

Tables (16)

Equations (2)

Applied Optics