Non-spherical single 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. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a cross-sectional view of a non-spherical single lens according to an embodiment of the present invention. 
     FIGS. 2A, 2B, 3A, 3B, 4A and 4B show various aberrations in respective embodiments of the non-spherical single lens according to the present invention. 
    
    
     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&#39; 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&#39; 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 Δ 1  indicates that is belongs to the first surface). This is because the object side of the first surface of radius r 1  is air (N=1) and the image side thereof is the material (N&#39;=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=˜ and therefore, by the radius of curvature of the first surface being r 1  =˜, 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 1 , coma II 1  and astigmatism III 1  created in the first surface are I 1  =II 1  =III 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 |t|=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 2  and astigmatism III 2  which are the off-axis aberrations in the second surface 3 are 
     
         II.sub.2 =III.sub.2 =0. 
    
     However, in the second surface 3, as can be seen from equation (1), spherical aberration I 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 1  is caused to be created in the first surface 2 and I 1  is rendered into I 1  =-I 2 , whereby spherical aberration is corrected. Even if a spherical surface is introduced into the first surface 2, the incidence height h 1  of the pupil paraxial ray is h 1  =0 and therefore, as can be seen from equations (2) and (3), coma II 1  and astigmatism III 1  created in the first surface can be made to remain to be of the value of II 1  =III 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 Δx from the spherical surface at a point whereat the y-coordinates are H is 
     
         Δx=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 2  in the second surface 3 by the spherical aberration I 1  in the first surface 2, the non-sphericity coefficient B of the first surface 2 is first set to 
     
         B=-I.sub.2 /8(n-1)f.sup.3. 
    
     Here, I 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 
     
         -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. 
     
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Embodiment 1                                                              
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f = 10                 NA = 0.25                                          
r1 = ∞           r2 = -8.0593                                       
d = 8.0593             n = 1.80593                                        
B = -7.78789 × 10.sup.-4                                            
C = -1.47206 × 10.sup.-5                                            
D = -3.17439 × 10.sup.-7                                            
E = -1.77517 × 10.sup.-8                                            
I = II = III = I* = II* = 0                                               
P = 0.55373                                                               
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Embodiment 2                                                              
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f = 10                 NA = 0.325                                         
r1 = ∞           r2 = -8.0593                                       
d = 8.0593             n = 1.80593                                        
B = -7.78789 × 10.sup.-4                                            
C = -1.47206 × 10.sup.-5                                            
D = -2.42841 × 10.sup.-7                                            
E = -2.57976 × 10.sup.-8                                            
I = II = III = I* = II** = 0                                              
P = 0.55373                                                               
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Embodiment 3                                                              
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f = 10                 NA = 0.35                                          
r1 = ∞           r2 = -8.0593                                       
d = 8.0593             n = 1.80593                                        
B = -7.78789 × 10.sup.-4                                            
C = -1.47206 × 10.sup.-5                                            
D = -1.73082 × 10.sup.-7                                            
E = -3.16602 × 10.sup.-8                                            
I = II = III = I* = II* = 0                                               
P = 0.55373                                                               
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