Non-spherical single lens

A non-spherical single lens comprises a first surface which is a non-spherical surface, the first surface being substantially zero in the curvature thereof near the optic axis and being of a non-spherical shape gradually displaced outwardly away from the optic axis, and a second surface which is a convex spherical surface of a radius of curvature r, .vertline.r.vertline. being a value equal or approximate to d which is the thickness of the lens.

BACKGROUND OF THE INVENTION 
1. Field of the Invention 
This invention relates to a lens used to condense or collimate 
substantially monochromatic light emitted from a laser or an LED, and in 
particular to a non-spherical single lens whose NA is up to the order of 
0.35 and in which correction of aberrations is effected over a relatively 
wide field of view. 
2. Related Background Art 
Non-spherical single lenses have heretofore been proposed in Japanese 
Laid-open patent application Nos. 64714/1982, 201210/1982, 1983 (U.S. 
counterpart is U.S. Pat. No. 4,415,238) and Japanese Laid-open patent 
application No. 68711/1983. However, any of these propositions is designed 
chiefly as an objective for an optical head. Accordingly, in these lenses, 
correction of aberrations is effected only in a very slight range on and 
near the axis, and where these are used, for example, as collimators, 
aberrations are created unless a light source is disposed at a very much 
limited position the focus position on the optic axis of the lens, and 
thus, great pains have often been bestowed on assembly, adjustment, etc. 
SUMMARY OF THE INVENTION 
It is an object of the present invention to provide a non-spherical single 
lens in which correction of aberrations is effected over a much wider 
field of view than in the conventional non-spherical single lenses, 
thereby alleviating the need for such great care in accuracy of assembly 
and adjustment. 
In the non-spherical single lens according to the present invention, where 
the single lens is used at a reduction magnification, a first surface as 
viewed from the object side from which a light beam enters the single lens 
toward the image side is a non-spherical surface and a second surface is a 
spherical surface of a radius of curvature r which is convex toward the 
image side, and said non-spherical surface is of such a non-spherical 
shape that the curvature thereof near the optic axis is substantially zero 
and the surface protrudes toward the object side away from the optic axis. 
That is, the shape of the non-spherical surface is a shape in which as a 
whole, a concave surface faces the object side, and the amount of 
displacement from a plane perpendicular to the optic axis increases away 
from the optic axis. Further, if the on-axis thickness of the lens is d, 
the absolute value of the radius of curvature r is equal or approximate to 
d, whereby there is obtained a non-spherical single lens which achieves 
the above object.

DESCRIPTION OF THE PREFERRED EMBODIMENTS 
FIG. 1 shows a cross-section of a non-spherical single lens according to an 
embodiment of the present invention, and also shows the optical path when 
the non-spherical single lens is used as a condensing lens in a device 
such as a compact disk or an optical disk. Accordingly, there is shown a 
case where the non-spherical single lens is used at a reduction 
magnification, and if the light beam incidence side of the lens is 
generally represented as the object side, the object side surface 2 of the 
lens is a non-spherical surface and the image side surface 3 of the lens 
is a spherical surface. In the case of the use converse to that shown in 
FIG. 1, that is, in a case where the non-spherical single lens is used at 
an enlargement magnification as the collimator of a semiconductor laser or 
the like, the spherical surface 3 is the object side surface and the 
non-spherical surface 2 is the image side surface. 
The non-spherical single lens according to the present invention, unlike a 
lens such as an objective for an optical head which is directed to the 
correction of only the aberrations on and near the axis and in which 
chiefly spherical aberration and coma need only be eliminated, requires 
that astigmatism also be eliminated in order that a good imaging 
performance may be maintained over a relatively wide field of view. 
The present invention will hereinafter be described in detail by the use of 
the method of discription described in detail in The Lens Design (written 
by Matsui and published by Kyoritsu Publishing Co., Ltd.). 
According to the description in the same book, the third-order aberration 
coefficients I, II, III, IV and V for a spherical surface can be expressed 
as: 
##EQU1## 
where h is the height of incidence of the light ray onto the surface, Q is 
the invariable amount of Abble, N and N' are the refractive indices of the 
medium, h and Q are the heights of incidence and the invariable amount of 
Abbe, respectively, regarding the principal ray, r is the radius of 
curvature of the spherical surface, and S and S' are the distances from 
the spherical surface to the object and the image, respectively. 
From the paraxial imaging formula 
##EQU2## 
and therefore 
##EQU3## 
shown above is expressed as 
##EQU4## 
If this is applied particularly to the first surface of the single lens, 
##EQU5## 
(where the suffix 1 of .DELTA..sub.1 indicates that is belongs to the 
first surface). This is because the object side of the first surface of 
radius r.sub.1 is air (N=1) and the image side thereof is the material 
(N'=n) of the single lens. 
Accordingly, when, as shown in FIG. 1, a substantially parallel light beam 
is incident on the single lens 1, S1=.about. and therefore, by the radius 
of curvature of the first surface being r.sub.1 =.about., there can be 
obtained 
##EQU6## 
That is, when the curvature of the first surface is zero, if a parallel 
light beam is incident on the first surface, from equations (1)-(3) above, 
the values of spherical aberration I.sub.1, coma II.sub.1 and astigmatism 
III.sub.1 created in the first surface are I.sub.1 =II.sub.1 =III.sub.1 =0 
(in the case of the incidence of a parallel light beam). 
Also, if the position of the pupil of the lens 1 is made coincident with 
the position of the first surface 2 and the radius of curvature r of the 
second surface 3 which is a convex surface facing the image side is set so 
as to be equal to the on-axis thickenss d of the lens, the pupil paraxial 
ray passing through the center of said pupil corresponds to a case where 
it is incident on the second surface from the center of curvature of the 
second surface 3 and therefore, this pupil paraxial ray is not refracted 
by the second surface 3, but emerges from the second surface 3 toward the 
image side. That is, by the distance t between the second surface 3 and 
the pupil being .vertline.t.vertline.=d (-t=d), 
##EQU7## 
and therefore, by -r being -r=d, r becomes r=t and thus, Q shown in 
equation (10) is Q0. It should be noted that r is a value including a 
sign, and the position of the center of curvature thereof lies on the 
first surface 1 and exists more adjacent to the object side than the 
surface 3 and therefore, r is a negative value. At this time, as is 
apparent from equations (2) and (3), coma II.sub.2 and astigmatism 
III.sub.2 which are the off-axis aberrations in the second surface 3 are 
EQU II.sub.2 =III.sub.2 =0. 
However, in the second surface 3, as can be seen from equation (1), 
spherical aberration I.sub.2 cannot be zero in the above-described 
construction, and therefore, a non-spherical surface is introduced into 
the first surface 2, spherical aberration I.sub.1 is caused to be created 
in the first surface 2 and I.sub.1 is rendered into I.sub.1 =-I.sub.2, 
whereby spherical aberration is corrected. Even if a spherical surface is 
introduced into the first surface 2, the incidence height h.sub.1 of the 
pupil paraxial ray is h.sub.1 =0 and therefore, as can be seen from 
equations (2) and (3), coma II.sub.1 and astigmatism III.sub.1 created in 
the first surface can be made to remain to be of the value of II.sub.1 
=III.sub.1 =0. 
A method of negating spherical aberration will now be described. 
When as a form representing the shape of the non-spherical surface which is 
the first surface 2, for the sake of convenience, the x-axis is chosen 
coincidently with the direction of travel of light on the optic axis and 
the y-axis is chosen so as to be orthogonal thereto and pass through the 
vertex of the first surface 2 (see FIG. 1), it is to be understood that 
the amount of displacement .DELTA.x from the spherical surface at a point 
whereat the y-coordinates are H is 
EQU .DELTA.x=BH.sup.4 +CH.sup.6 DH.sup.8 +EH.sup.10, 
where B, C, D and E are non-sphericity coefficients representative of 
degrees of non-sphericity. In order to negate the spherical aberration 
I.sub.2 in the second surface 3 by the spherical aberration I.sub.1 in the 
first surface 2, the non-sphericity coefficient B of the first surface 2 
is first set to 
EQU B=-I.sub.2 /8(n-1)f.sup.3. 
Here, I.sub.2 is generally a positive value and therefore, B is a negative 
value and by this, it is seen that the shape of the non-spherical surface 
which is the first surface 2 must be a shape displaced toward the object 
side. Next, the non-sphericity coefficients C, D and E are determined so 
as to eliminate any residual spherical aberration. Thereby, a lens in 
which spherical aberration, coma and astigmatism are corrected can be 
embodied in practice. 
In the foregoing description, it has been shown that spherical aberration, 
coma and astigmatism can be completely eliminated if the value r of the 
radius of curvature of the second surface 3 is made coincident with the 
on-axis thickness d of the lens 1, whereas these aberrations need not 
always be completely eliminated, but if the relation between d and r is 
within the range of value of 
EQU -1.1d&lt;r&lt;-0.9d, 
there will be obtained a lens in which aberrations are sufficiently 
corrected in practical use. As can be seen from equation (10), if the 
relation between r and d departs from this range, correction of 
aberrations, particularly, coma, will become difficult and it will be 
difficult to obtain a single lens usable at a wide angle of view. 
Three embodiments of the non-spherical single lens according to the present 
invention will be shown below. In the lists below, f is the focal length 
of the lens, NA is the numerical aperture, r1 is the radius of curvature 
of the first surface near the optic axis, r2 is the radius of curvature of 
the second surface, d is the on-axis thickness of the lens, and n is the 
refractive index of the lens. Further, B, C, D and E are the 
non-sphericity coefficients of the first surface, I is spherical 
aberration, II is coma, III is astigmatism, I is annular spherical 
aberration, II is annular coma, and P is Petzval sum. The vertical 
aberration in Embodiment 1 is shown in FIG. 2A, the lateral aberration in 
Embodiment 1 is shown in FIG. 2B, the vertical aberration in Embodiment 2 
is shown in FIG. 3A, the lateral aberration in Embodiment 2 is shown in 
FIG. 3B, the vertical aberration in Embodiment 3 is shown in FIG. 4A, and 
the lateral aberration in Embodiment 3 is shown in FIG. 4B. In FIGS. 2B, 
3B and 4B, solid lines (meridional) are approximately symmetrical about 
the origin and dotted lines (sagittal) are perfectly symmetrical. 
______________________________________ 
Embodiment 1 
______________________________________ 
f = 10 NA = 0.25 
r1 = .infin. r2 = -8.0593 
d = 8.0593 n = 1.80593 
B = -7.78789 .times. 10.sup.-4 
C = -1.47206 .times. 10.sup.-5 
D = -3.17439 .times. 10.sup.-7 
E = -1.77517 .times. 10.sup.-8 
I = II = III = I* = II* = 0 
P = 0.55373 
______________________________________ 
______________________________________ 
Embodiment 2 
______________________________________ 
f = 10 NA = 0.325 
r1 = .infin. r2 = -8.0593 
d = 8.0593 n = 1.80593 
B = -7.78789 .times. 10.sup.-4 
C = -1.47206 .times. 10.sup.-5 
D = -2.42841 .times. 10.sup.-7 
E = -2.57976 .times. 10.sup.-8 
I = II = III = I* = II** = 0 
P = 0.55373 
______________________________________ 
______________________________________ 
Embodiment 3 
______________________________________ 
f = 10 NA = 0.35 
r1 = .infin. r2 = -8.0593 
d = 8.0593 n = 1.80593 
B = -7.78789 .times. 10.sup.-4 
C = -1.47206 .times. 10.sup.-5 
D = -1.73082 .times. 10.sup.-7 
E = -3.16602 .times. 10.sup.-8 
I = II = III = I* = II* = 0 
P = 0.55373 
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