Spherical aberration correction and fabrication tolerances in geodesic lenses

Gary E. Betts; Gary E. Merx

doi:10.1364/AO.17.003969

Applied Optics
Vol. 17,
Issue 24,
pp. 3969-3974
(1978)
•https://doi.org/10.1364/AO.17.003969

Spherical aberration correction and fabrication tolerances in geodesic lenses

Gary E. Betts and Gary E. Merx

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Gary E. Betts and Gary E. Merx, "Spherical aberration correction and fabrication tolerances in geodesic lenses," Appl. Opt. 17, 3969-3974 (1978)

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Abstract

This paper describes a theoretical design of an integrated optical geodesic lens using a noncircular overlay to correct spherical aberrations. The corrected lens design can achieve diffraction-limited performance for an on-axis beam with an aperture equal to 90% of the lens diameter. The paper also contains an analysis of the tolerances, which must be met during the fabrication of such a lens to maintain a diffraction-limited spot and prescribed focal length. This analysis showed that the most sensitive parameter is the thickness of the overlay.

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	θ	N_rel	a_o	F_‡‡	Focal Length	Aperture	Too Deep⁺		Wrong Edge Rounding^*	X-Alignment		Y-Alignment⁺
	θ	N_rel	a_o	F_‡‡	Focal Length	Aperture	Size /	Location	Size	Size /	Location	Size /	Location
A	15°	1.01	− 1.45	6.5	11.50	90%	.8	− 60	0 – .15	.02	.08	.26	− 15
B	15°	1.02	− 1.2	5.3	9.56	90%	.7	− 41	0 – .1	.026	.03	.1	− 9
C	30°	1.02	− 2.2	1.8	3.19	90%	.25	− 8	0 – .13	.05	.42	.2	− 4
D	30°	1.04	− 1.4	1.7	2.99	90%	.28	− 7	0 – 13	.05	.31	.17	− 3.2
E	30°	1.08	− 1.1	1.4	2.56	90%	.24	− 4.5	0 – .1	.08	.20	.16	− 2.0
F	40°	1.05	− 1.5	1.0	1.85	90%	.14	− 3.2	0 – .08	.05	.5	.16	− 1.8
G	30°	1.04	− 2.27	1.3	2.32	90%	.17	− 4	0 – .07	.06	.3	.23	− 3
H	30°	1.04	− 1.002	2.0	3.70	90%	.4	− 11	0 – .19	.06	.3	.35	− 4
I	30°	1.04	− 1.4	2.1	2.99	70%	.08	− 5.5	0 – .02	.016	.65	.06	− 2.3
J	30°	1.04	− 1.4	3.0	2.99	50%	.02	− 5	0 – .004	.004	.75	.06	− 2.3

Lens					Error

	Too Long⁺		Too Wide⁺		Bulge		Output EdgeRadius (R_o)⁺		N_rel^*⁺
	Size	Location	Size	Location	Size	Location	Size	Location	Size	Location R_L)
									(×10⁻⁴ R_L)
A	.08	−2.3	.4	15	.5	4.5	.1	3.7	9.7	−587
B	.13	−3.5	.19	10	.66	5	.17	5	4.1	−281
C	.05	−.35	.2	3	.4	.4	.03	.4	2.8	−60
D	.09	−.6	.3	4	.4	.7	.07	.8	1.5	−31
E	.14	−1	.12	2	.7	1.4	.07	.9	.39	−16
F	.06	−.3	.2	1.5	.35	.6	.04	.3	.9	−12
G	.06	−.3	.3	2	.55	1	.04	.4	.18	−28
H	.1	−.9	.4	.35	.50	.6	.1	1.4	2.9	−30
	.09	−.6	.09	2	.35	1.4	.016	.46	.77	−24
J	.08	−.85	.06	1.9	.3	.007	.004	.36	.25	−19


Lens	Depth (μm)	Edge Rounding^* (μm)	X align (μm)	Y align (μm)	Too Long (μm)	Too Wide (μm)	Bulge (μm)	Exit Radius (μm)
A	18	10	39	.69	3.9	.63	1.2	2.6
B	.22	10	35	1.1	2.2	.9	.9	1.6
C	.40	10	7.5	.77	7.2	.96	2	7.2
D	.42	10	7.0	.88	3.7	.67	1.7	3.2
E	.52	10	6.5	1.1	1.9	1.2	.84	2.4
F	.62	10	3.6	1.0	5.0	1.1	1.6	6.4
G	.61	10	6.3	.73	6.3	.93	1.0	5.3
H	.32	10	7.7	.75	3.1	2.1	1.5	2.3
I	.73	20	5.9	1.6	4.3	1.8	1.9	8.6
J	1.2	30	8.0	2.3	4.6	2.8	1.3	16

	Lens				Radius Error
θ	N_rel	$\frac{R_{O}}{R_{L}}$	Aperture	Size	Location	Allowed for 1 μm Total Size Increase
20°	1.057	1.05	40%	.015	1.5	2.2 μm
23°	1.062	1.00	40%	.016	1.3	3.0 μm
25°	1.085	1.05	50%	.035	1.0	3.0 μm

Lens	N_rel	Point on Dispersion Curve Used	N_rel Error Causing 1μm Spot Size Increase	Slope of Dispersion Curve^*	Thickness Change Causing 1μm change in Focal Length	Thickness Change Causing 1μm Size Increase Neglecting Focal Length Change	Thickness Error Causing 1μm Size Increase, both Causes Considered
A	1.01	a	3.6 × 10⁻⁶	.014 μm⁻¹	0.12 Å	15 Å	2.6 Å
B	1.02	b	6.6 × 10⁻⁶	.0015 μm⁻¹	2.3 Å	330 Å	44 Å
C	1.02	b	1.0 × 10⁻⁵	.0015 μm⁻¹	11.0 Å	480 Å	67 Å
D	1.04	c	1.8 × 10⁻⁵	.16 μm⁻¹	0.20 Å	8.2 Å	1.1 Å
J	1.04	c	6.1 × 10⁻⁵	.16 μm⁻¹	0.33 Å	50 Å	3.8 Å
E	1.08	d	3.2 × 10⁻⁵	.014 μm⁻¹	4.5 Å	360 Å	23 Å

Abstract

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Equations (12)

Applied Optics