Abstract:
A graded refractive index single lens system comprising at least one surface formed spherically with the radius of curvature being large wherein N.A. is large and various aberrations including off-axial aberrations are well-corrected.

Description:
BACKGROUND OF THE INVENTION 
     (a) Field of the Invention 
     The present invention relates to a lens system using inhomogeneous material, especially a graded refractive index (GRIN) single lens system used as an objective lens system for optical video disks, etc. 
     (b) Description of the Prior Art 
     Recently there have developed apparatuses which read, by the converging of a laser beam to a microspot, the information which is recorded with high density on recording medium such as optical video disks, digital audio disks, etc. 
     In such apparatuses, it is necessary for an objective lens system used for the recording and the reproducing of information to be compact and light because the objective lens system is driven directly for the purpose of autofocusing and auto-tracking. It is also necessary for the objective lens system to have a large N.A. in order to obtain a smaller spot size of a laser beam which is converged on a recording medium. 
     As such an objective lens system, a combination of a plurality of homogeneous spherical lenses or a single homogeneous aspherical lens, especially for the purposes of being compact and light, has hitherto been in use. 
     Moreover, besides these homogeneous lenses, a GRIN single lens system using inhomogeneous material for economy of manufacture, compactness, and light weight has been known recently. 
     In the early GRIN lens system, only the correction of spherical aberration was considered. 
     As is well known, it is necessary for an objective lens system used for optical video disks, etc. to have aberrations well-corrected in the range of diameters of 0.1-0.2 mm on the disk surface and, therefore, not ony spherical aberration but also coma should be well-corrected. 
     A GRIN single lens system disclosed, for example, in Japanese Published Unexamined Patent Application No. 6354/80 has at least one of the refracting surfaces thereof formed as a spherical surface. In this lens system, spherical aberration is well-corrected, but the correction of other aberrations is not sufficient. 
     GRIN single lens systems intended to correct off-axial aberrations, especially coma are disclosed in Japanese Published Unexamined Patent Application Nos. 122512/83 and 62815/84. In these lens systems, the refracting surface of the GRIN lens is formed as spherical one, and the radius of curvature of this spherical surface and the higher-order coefficients of the refractive index distibution are arranged  ranged so that both spherical aberration and coma can be corrected. However, in the former lens system (Japanese Published Unexamined Patent Application No. 6354/80), it cannot be said that the correction of aberrations thereof is sufficient. The latter lens systems (Japanene Published Unexamined Patent Application Nos. 122512/83 and 62815/84) have defects in that, for example, the shapes of these lenses have such strongly meniscus shapes that the manufacture thereof is difficult. 
     SUMMARY OF THE INVENTION 
     It is a primary object of the present invention to provide a graded refractive index (GRIN) single lens system with a comparatively large N.A. comprising at least one surface thereof formed spherically, the radius of curvature of which is large so that the manufacture thereof is easy. 
     In the GRIN single lens system according to the present invention, the refractive index distribution is cylindrically symmetric to the optical axis and is expressed by the following formula: 
     
         n.sup.2 =n.sub.0.sup.2 [1-(gr).sup.2 +h.sub.4 (gr).sup.4 +h.sub.6 (gr).sup.6 + . . . ] 
    
     where n 0  represents the refractive index on the optical axis of the lens, r represents the radial distance from the optical axis, g is the parameter representing the degree of gradient of the refractive index distribution, and, h 4  and h 6  respectively represent the 4th- and 6th-order coefficients of the refractive index distribution. 
     In the GRIN lens whose refractive index distribution is expressed by the above mentioned formula, in order to make the refracting power of the entire lens system at a predetermined value and to correct aberrations, both the refracting power of the refracting surface and that of the lens medium are appropriately arranged. The refracting power of the lens medium can be determined precisely from the description in Journal of Optical Society of America Vol. 61, No. 7, pp. 879-885, &#34;Inhomogeneous Lens III, Paraxial Optics&#34;. According to this description, when gD is less than π/2 (D represents the thickness of the lens), the magnitude of the refractive power of the lens medium can be known by the value of gD. The degree of the gradient of the refractive index distribution is expressed by the parameter g. The value of g is also influenced by the shape of the lens, for example, the diameter thereof. In an ordinary homogeneous lens, the lens whose focal length, radii of curvatures, length and diameter are multiplied by a fixed number compared with those of an original lens is optically equivalent to the original lens. In a GRIN lens, only when the parameter g is multiplied by the reciprocal of that fixed number compared with that of an original lens in addition to the values obtained as above, can the lens optically equivalent to the original lens be obtained. Therefore, the gradient of the refractive index distribution can be arranged not only by the shape of the lens but also by the value of gφ (where φ represents the diameter of the lens) of gf (where f represents the focal length of the lens). 
     Thus, in a GRIN lens, when the values of some parameters of D, gD, gφ, gf, etc. are suitably chosen, and when some parameters of them are suitably selected, these selected parameters are correlated with one another and suitable values are fixed, it is possible to obtain the GRIN lens wherein N.A. is large and aberrations are well-corrected though the radius of curvature of the lens surface is kept large. 
     The sets of parameters to be selected may be as follows: 
     (a) parameters D, gD, gφ 
     (b) parameters D, gf 
     (c) parameters D, gD, gf 
     (d) parameters D, (g-0.5)D 
     When the set (a) is selected from the above mentioned sets, it is desirable for the parameters of this set to satisfy the following conditions (1), (2), and (3): 
     
         gD&lt;0.51                                                    (1) 
    
     
         0.3&lt;gφ&lt;0.7                                             (2) 
    
     
         0.28f&lt;D                                                    (3) 
    
     The conditions (1) and (2) define the refracting power of the lens medium and the degree of the gradient of the refractive index distribution, and are established in order to make the radius of curvature of a refracting surface large when aberrations are to be well-corrected. 
     If the value of gD becomes large, spherical aberration and coma generated by the ray passing through the lens medium will become large. In order to correct these aberrations, it is necessary to make the refracting surface at the opposite side of the long conjugate point a strong concave surface. If the value of gD under the condition (1) exceeds the limit of this condition, the radius of curvature of this surface will become strong so that the manufacture of the lens will become difficult. 
     Even if gD is within the limit of the condition (1), in case gφ exceeds the upper limit of the condition (2), the gradient of the refractive index distribution will become steep and the ray will be curved largely in the lens medium. In this state, it is necessary for the good correction of aberrations to make the refracting surface at the opposite side of the long conjugate point a strong concave surface, which is against the object of the present invention. 
     The lower limit of the condition (2) and the condition (3) itself are established in order to correct the various aberrations well. If the value of gφ under the condition (2) exceeds the lower limit of this condition or the condition (3) is not satisfied, it will be impossible to correct both spherical aberration and coma. 
     When the set (b) of the parameters is selected, it is necessary to satisfy the following conditions (4) and (5): 
     
         0.96 f&lt;D&lt;1.536 f                                           (4) 
    
     
         0.63&lt;gf                                                    (5) 
    
     The lower limit of the condition (4) and the condition (5) itself are established in order to correct both astigmatism and sign condition with good balance and, furthermore, keep the sign condition in a good state. 
     If the value of D under the condition (4) exceeds the lower limit thereof, astigmatism will deteriorate. If the condition (5) is not satisfied, the sign condition will deteriorate. 
     When these conditions are satisfied, it will be possible to correct the sign condition easily while astigmatism is kept in a good state. 
     The upper limit of the condition (4) is established in order to keep the minimum necessary working distance provided that the condition (5) is satisifed. In other words, if the value of D under the condition (4) exceeds the upper limit thereof, it will be impossible to keep the necessary working distance. 
     When the set (c) of the parameters is selected, it is necessary to satisfy the following conditions (6), (7) and (8): 
     
         D&lt;1.08 f                                                   (6) 
    
     
         gf&lt;0.604                                                   (7) 
    
     
         0.51&lt;gD                                                    (8) 
    
     The condition (6) defines the length D of the lens and is established in order to keep the necessary working distance while maintaining balance with astigmatism. If this condition is not satisfied, it will be impossible to keep a sufficient working distance. 
     The condition (7) is established in order to correct astigmatism. When this condition is not satisfied even if the value of D is chosen to satisfy the condition (6), astigmatism will deteriorate and it will be impossible to make N.A. large. 
     The upper limit of the refracting power of the lens medium is determined to satisfy the conditions (6) and (7). But in order to correct various aberrations with good balance, it is more desirable to have the refracting power distributed suitably between the power of the refracting surface and that of the lens medium. If the refracting power of the lens medium exceeds a limit value and becomes small, astigmatism will deteriorate remarkably and it will be very hard to correct it under the condition wherein astigmatism is balanced with spherical aberration. 
     The condition (8) defines the lower limit of the refracting power of the lens medium. If the value of gD under the condition (8) exceeds the lower limit thereof, it will be impossible to make N.A. large. 
     In the GRIN lens satisfying the conditions (6), (7) and (8), it will be possible to correct spherical aberration and astigmatism with good balance when the radius R 1  of curvature of the refracting surface at the long conjugate side and the radius R 2  of that at the short conjugate side satisfy the following condition (9) so that the refracting powers of both surfaces are well balanced: 
     
         2&lt;|R.sub.2 /R.sub.1 |                    (9) 
    
     When the 4th-order coefficient h 4  of the refractive index distribution satisfy the following condition (10), it will be possible to keep a spherical aberration curve in a good shape and maintain the root mean square of wave aberration at a very small value of λ/40 near the optical axis. 
     
         h.sub.4 &lt;0                                                 (10) 
    
     When the set (c) of the parameters, i.e., D, gf, gD is selected, even if the length D of the lens is longer to some extent, it will be possible to form the GRIN lens which also attains the object of the present invention provided that the value of gf is appropriately arranged. In other words, the following condition (11), (12) and (13) should be satisfied. 
     
         1.152 f&lt;D&lt;1.392 f                                          (11) 
    
     
         gf&lt;0.562                                                   (12) 
    
     
         0.51&lt;gD                                                    (13) 
    
     The lower limit of the condition (11) and the upper limit of the condition (12) are established in order to correct astigmatism. If these limits are exceeded, it will be impossible to correct astigmatism sufficiently when N.A. is to be large. 
     The upper limit of the condition (11) is established in order to make the radius of curvature of at least one refracting surface large. If the value of D under the condition (11) exceeds the upper limit thereof, it will be impossible to make the radius of curvature of the refracting surface large. 
     The condition (13) is established for the same reason as for the establishment of the condition (8). 
     Finally, when the set (d) of the parameters is selected, it will be necessary to satisfy the following conditions (14) and (15). 
     
         1.54 f &lt;D                                                  (14) 
    
     
         -4&lt;(g-0.5)D                                                (15) 
    
     The condition (14) relates to the length of the lens and is correct astigmatism. If this condition is not satisfied, it will be impossible to correct astigmatism when N.A. is enlarged to about 0.7. 
     The condition (15) relates to the refracting power of the lens medium. As is mentioned above, in a GRIN lens, the refracting power of the entire lens can be divided between the refracting power of the refracting surface and that of the lens medium, the balance of which is important when the various aberrations are to be corrected. For keeping a good balance it is necessary to take the length of the lens itself into account. If this condition (15) is not satisfied, spherical aberration will deteriorate. Depending on the refracting power of the refracting surface, it may be hard to keep a good balance of aberrations and impossible to correct them. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 shows a sectional view of Embodiment 1 of the GRIN single lens system according to the present invention; 
     FIG. 2 shows a sectional view of Embodiments 2, 3, 4, 20 and 21 of the GRIN single lens system according to the present invention; 
     FIG. 3 shows a sectional view of Embodiments 5 through 19 of the GRIN single lens system according to the present invention; 
     FIG. 4 shows a sectional view of Embodiments 22, 23, 24, 25, 27 and 31 of the GRIN single lens system according to the present invention; 
     FIG. 5 shows a sectional view of Embodiments 26, 28, 29, 30 and 32 of the GRIN single lens system according to the present invention; 
     FIG. 6 shows a sectional view of Embodiments 33, 34, 40 and 41 of the GRIN single lens system according to the present invention; 
     FIG. 7 shows a sectional view of Embodiments 35, 36, 37, 38, 39 and 42 of the GRIN single lens system according to the present invention; 
     FIG. 8 shows a sectional view of Embodiment 44 of the GRIN single lens system according to the present invention; 
     FIG. 9 shows a sectional view of Embodiments 43, 45, 46, 47, 48, 49, 50 and 51 of the GRIN single lens system according to the present invention; 
     FIG. 10 shows a sectional view of Embodiments 52, 53, 54, 55, 56, 58, 59, 60 and 62 of the GRIN single lens system according to the present invention; 
     FIG. 11 shows a sectional view of Embodiments 57, 61, 63 and 64 of the GRIN single lens system according to the present invention; and 
     FIGS. 12 through 75, respectively, show graphs illustrating aberration curves of Embodiment 1 through 64 of the GRIN lens system according to the present invention. 
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     Preferred Embodiments of the GRIN lens system according to the present invention as described above are explained below. 
     Embodiments 1 through 21 according to the present invention shown in the following numerical data have the features in the parameters gD, gφ, D defined and satisfy the conditions (1), (2) and (3). 
     
         ______________________________________Embodiment 1R.sub.1 = 2.847     R.sub.2 = ∞                  D = 2.832   n.sub.0 = 1.65g = 0.12  h.sub.4 = 2.404                  h.sub.6 = 47.99                              φ = 3.2f = 3.5556     NA = 0.45    WD = 1.21   gD = 0.340gφ = 0.384     D/f = 0.796Embodiment 2R.sub.1 = 3.009     R.sub.2 = -62.404                  D = 3.333   n.sub.0 = 1.65g = 0.12  h.sub.4 = 1.716                  h.sub.6 = 32.20                              φ = 3.2f = 3.5556     NA = 0.45    WD = 0.99   gD = 0.400gφ = 0.384     D/f = 0.937Embodiment 3R.sub.1 = 3.194     R2 = -24.832 D = 3.750   n.sub.0 = 1.65g = 0.12  h.sub.4 = 1.298                  h.sub.6 = 23.65                              φ = 3.2f =  3.5556     NA = 0.45    WD = 0.84   gD = 0.45gφ = 0.384     D/f = 1.055Embodiment 4R.sub.1 = 3.458     R.sub.2 = -12.852                  D = 4.167   n.sub.0 = 1.65g = 0.12  h.sub.4 = 1.082                  h.sub.6 = 18.89                              φ = 3.2f = 3.5556     NA = 0.45    WD = 0.73   gD = 0.5gφ = 0.384     D/f = 1.172Embodiment 5R.sub.1 = 2.649     R.sub.2 = 60.550                  D = 1.333   n.sub.0 = 1.65g = 0.15  h.sub.4 = -0.597                  h.sub.6 = -2.040                              φ = 3.2f = 3.5556     NA = 0.45    WD = 2.01   gD = 0.2gφ = 0.48     D/f = 0.375Embodiment 6R.sub.1 = 2.716     R.sub.2 = 40.040                  D = 1.667   n.sub.0 = 1.65g = 0.15  h.sub.4  = -0.611                  h.sub.6 = -0.923                              φ = 3.2f = 3.5556     NA = 0.45    WD = 1.82   gD = 0.25gφ = 0.48     D/f = 0.469Embodiment 7R.sub.1 = 2.792     R.sub.2 = 30.725                  D = 2.0     n.sub.0 = 1.65g = 0.15  h.sub.4 = -0.646                  h.sub.6 = -0.816                              φ = 3.2f = 3.5556     NA = 0.45    WD = 1.64   gD = 0.3gφ = 0.48     D/f = 0.562Embodiment 8R.sub.1 = 2.877     R.sub.2 = 25.630                  D = 2.333   n.sub.0 = 1.65g = 0.15  h.sub.4 = -0.691                  h.sub.6 = -0.993                              φ = 3.2f = 3.5556     NA = 0.45    WD = 1.45   gD = 0.35gφ = 0.48     D/f = 0.656Embodiment 9R.sub.1 = 2.973     R.sub.2 =  22.668                  D = 2.667   n.sub.0 = 1.65g = 0.15  h.sub.4 = -0.735                  h.sub.6 = -1.227                              φ = 3.2f = 3.5556     NA = 0.45    WD = 1.28   gD = 0.4gφ = 0.48     D/f = 0.750Embodiment 10R.sub.1 = 3.081     R.sub.2 = 21.061                  D = 3.0     n.sub.0 = 1.65g = 0.15  h.sub.4 = -0.774                  h.sub.6 = -1.420                              φ = 3.2f = 3.5556     NA = 0.45    WD = 1.11   gD = 0.45gφ = 0.48     D/f = 0.844Embodiment 11R.sub.1 = 3.206     R.sub.2 = 20.562                  D = 3.333   n.sub.0 = 1.65g = 0.15  h.sub.4 = -0.802                  h.sub.6 = -1.526                              φ = 3.2f = 3.5556     NA = 0.45    WD = 0.95   gD = 0.5gφ =  0.48     D/f = 0.937Embodiment 12R.sub.1 = 2.916     R.sub.2 = 17.811                  D = 2.059   n.sub.0 = 1.65g = 0.17  h.sub.4 = -1.247                  h.sub.6 = -5.704                              φ = 3.2f = 3.5556     NA = 0.45    WD = 1.59   gD = 0.35gφ = 0.544     D/f = 0.579Embodiment 13R.sub.1 = 3.009     R.sub.2 = 14.981                  D = 2.353   n.sub.0 = 1.65g = 0.17  h.sub.4 = -1.163                  h.sub.6 = -4.370                              φ = 3.2f = 3.5556     NA = 0.45    WD = 1.44   gD = 0.4gφ = 0.544     D/f = 0.662Embodiment 14R.sub.1 = 3.110     R.sub.2 = 12.932                  D = 2.647   n.sub.0 = 1.65g = 0.17  h.sub.4 = -1.099                  h.sub.6 = -3.524                              φ = 3.2f = 3.5556     NA = 0.45    WD = 1.28   gD = 0.45gφ = 0.544     D/f = 0.744Embodiment 15R.sub.1 = 3.219     R.sub.2 = 11.371                  D = 2.941   n.sub.0 = 1.65g = 0.17  h.sub.4 = -1.047                  h.sub.6 = -2.937                              φ = 3.2f = 3.5556     NA = 0.45    WD = 1.12   gD = 0.5gφ = 0.544     D/f = 0.827Embodiment 16R.sub.1 = 3.206     R.sub.2 = 9.178                  D = 2.250   n.sub.0 = 1.65g = 0.20  h.sub.4 = -1.092                  h.sub.6 = -2.748                              φ = 3.2f = 3.5556     NA = 0.45    WD = 1.48   gD = 0.45gφ = 0.64     D/f = 0.633Embodiment 17R.sub.1 = 3.318     R.sub.2 = 7.883                  D = 2.5     n.sub.0 = 1.65g = 0.20  h.sub.4 = -0.986                  h.sub.6 = -2.099                              φ = 3.2f = 3.5556     NA = 0.45    WD = 1.34   gD = 0.5gφ = 0.64     D/f = 0.703Embodiment 18R.sub.1 = 3.148     R.sub.2 = 65.733                  D = 3.5     n.sub.0 = 1.7g = 0.12  h.sub.4 = 0.909                  h.sub.6 = 19.14                              φ = 3.2f = 3.5556     NA = 0.45    WD = 0.89   gD = 0.42gφ = 0.384     D/f = 0.984Embodiment 19R.sub.1 = 3.149     R.sub.2 = 10.286                  D = 2.5     n.sub.0 = 1.7g = 0.17  h.sub.4 = -1.169                  h.sub.6 = -3.832                              φ = 3.2f = 3.5556     NA = 0.45    WD = 1.34   gD = 0.425gφ = 0.544     D/f = 0.703Embodiment 20R.sub.1 = 2.760     R.sub.2  = -14.835                  D = 3.0     n.sub.0 = 1.55g = 0.13  h.sub.4 = 1.839                  h.sub.6 = 36.50                              φ = 3.2f = 3.5556     NA = 0.45    WD = 1.18   gD = 0.39gφ = 0.416     D/f = 0.844Embodiment 21R.sub.1 = 2.512     R.sub.2 = -43.624                  D = 1.5     n.sub.0 = 1.55g = 0.16  h.sub.4 = -0.608                  h.sub.6 = -2.217                              φ = 3.2f = 3.5556     NA = 0.45    WD = 1.93   gD = 0.24gφ = 0.512     D/f = 0.422______________________________________ 
    
     where R 1 , R 2  respectively represent the radii of curvatures of the lens surfaces, D represents the length of the lens, n 0  represents the refractive index on the optical axis of the lens, the parameter g represents the gradient of the refractive index distribution, h 4  and h 6  respectively represent the 4th- and 6th-order coefficients of the refractive index distribution, φ represents the diameter of the lens, f represents the focal length of the lens, NA represents the numerical aperture at the side of the disk, and WD represents the distance between the lens and the disk. Both the coefficient g of the refractive index distribution and the value of refractive index are for the wave length λ=780 nm. 
     Embodiments 1 through 21 according to the present invention respectively satisfy the conditions (1) through (3). 
     Embodiment 1 according to the present invention is shown in FIG. 1 and is a plano-convex lens having a convex surface at a light source side not shown in this figure (at a long conjugate side). Each of Embodiments 2, 3, 4, 20 and 21 according to the present invention is, as shown in FIG. 2, a biconvex lens having a stronger convex surface at a light source side. Each of Embodiments 5 through 19 is, as shown in FIG. 3, a positive meniscus lens having a convex surface at a light source side. 
     Embodiments 22 through 32 according to the present invention shown in the following numerical data are the lenses satisfying the conditions (4) and (5). 
     
         ______________________________________Embodiment 22R.sub.1 = 1.234     R.sub.2 = 7.004                  D = 1.440   n.sub.0 = 1.50g = 0.646 h.sub.4 = -0.500                  h.sub.6 = -0.426                              f = 1.0NA = 0.5Embodiment 23R.sub.1 = 1.013     R.sub.2 = 4.115                  D = 1.140   n.sub.0 = 1.50g = 0.667 h.sub.4 = -0.718                  h.sub.6 = -1.144                              f = 1.0NA = 0.5Embodiment 24R.sub.1 = 1.167     R.sub.2 = 1.230                  D = 1.320   n.sub.0 = 1.50g = 0.708 h.sub.4 = -0.530                  h.sub.6 = -0.501                              f = 1.0NA = 0.5Embodiment 25R.sub.1 = 1.039     R.sub.2 = 1.719                  D = 1.080   n.sub.0 = 1.50g = 0.729 h.sub.4 = -0.640                  h.sub.6 = -0.809                              f = 1.0NA = 0.5Embodiment 26R.sub.1 = 1.190     R.sub.2 = 0.699                  D = 1.200   n.sub.0 = 1.50g = 0.792 h.sub.4 = -0.385                  h.sub.6 = -0.221                              f = 1.0NA = 0.5Embodiment 27R.sub.1 = 1.025     R.sub.2 = 1.462                  D = 1.080   n.sub.0 = 1.65g = 0.646 h.sub.4 = -0.847                  h.sub.6 = -1.547                              f = 1.0NA = 0.5Embodiment 28R.sub.1 = 1.244     R.sub.2 = 0.630                  D = 1.368   n.sub.0 = 1.65g = 0.688 h.sub.4 = -0.533                  h.sub.6 = -0.501                              f = 1.0NA = 0.5Embodiment 29R.sub.1 = 1.117     R.sub.2 = 0.874                  D = 1.140   n.sub.0 = 1.65g = 0.708 h.sub.4 =  -0.628                  h.sub.6 = -0.719                              f = 1.0NA = 0.5Embodiment 30R.sub.1 = 1.103     R.sub.2 = 0.902                  D = 1.020   n.sub.0 = 1.65g = 0.750 h.sub.4 = -0.570                  h.sub.6 = -0.568                              f = 1.0NA = 0.5Embodiment 31R.sub.1 = 1.231     R.sub.2 = 2.313                  D = 1.440   n.sub.0 = 1.50g = 0.667 h.sub.4 = -0.510                  h.sub.6 = -0.489                              f = 1.0NA = 0.6Embodiment 32R.sub.1 = 1.213     R.sub.2 = 1.065                  D = 1.380   n.sub.0 = 1.50g = 0.708 h.sub.4 = -0.488                  h.sub.6 = -0.432                              f = 1.0NA = 0.6______________________________________ 
    
     where R 1 , R 2  respectively represent the radii of curvatures of the lens surfaces, D represents the length of the lens, n 0  represents the refractive index on the optical axis of the lens, the parameter g represents the gradient of the refractive index distribution, h 4  and h 6  respectively represent the 4th- and 6th-order coefficients of the refractive index distribution, f represents the focal length of the lens, and NA represents the numerical aperture at the side of the disk. Both the coefficient g of the refractive index distribution and the value of refractive index are for the wave length λ=780 nm. 
     Among them, each of Embodiments 22 through 25, 27 and 31 according to the present invention is a positive meniscus lens as shown in FIG. 4. Each of Embodiments 26, 28 through 30 and 32 according to the present invention is a negative meniscus lens as shown in FIG. 5. 
     Embodiments 33 through 42 according to the present invention shown in the following numerical data are the lenses satisfying the conditions (6) through (8), and also satisfying the conditions (9) and (10). 
     
         ______________________________________Embodiment 33R.sub.1 = 0.866     R.sub.2 = -4.095                  D = 1.02    n.sub.0 = 1.5g = 0.563 h.sub.4 = -0.468                  h.sub.6 = 0.822                              f = 1.0NA = 0.5  WD = 0.283Embodiment 34R.sub.1 = 0.815     R.sub.2 = -9.840                  D = 0.876   n.sub.0 = 1.5g = 0.592 h.sub.4 = -0.704                  h.sub.6 = -1.402                              f = 1.0NA = 0.5  WD = 0.340Embodiment 35R.sub.1 = 0.903     R.sub.2 = 5.311                  D = 1.010   n.sub.0 = 1.65g = 0.521 h.sub.4 = -0.699                  h.sub.6 = -0.666                              f = 1.0NA = 0.5  WD = 0.259Embodiment 36R.sub.1 = 0.966     R.sub.2 = 2.452                  D = 1.060   n.sub.0 = 1.65g = 0.585 h.sub.4 = -0.936                  h.sub.6 = -2.223                              f = 1.0NA = 0.5  WD = 0.223Embodiment 37R.sub.1 = 0.905     R.sub.2 = 20.534                  D = 1.070   n.sub.0 = 1.65g = 0.479 h.sub.4 = -0.210                  h.sub.6 = 4.469                              f = 1.0NA = 0.5  WD = 0.240Embodiment 38R.sub.1 = 0.932     R.sub.2 = 3.942                  D = 1.050   n.sub.0 = 1.65g = 0.542 h.sub.4 = -0.846                  h.sub.6 = -1.904                              f = 1.0NA = 0.55 WD = 0.237Embodiment 39R.sub.1 = 0.891     R.sub.2 = 3.727                  D = 0.924   n.sub.0 = 1.65g = 0.554 h.sub.4 = -0.886                  h.sub.6 = -2.415                              f = 1.0NA = 0.55 WD = 0.295Embodiment 40R.sub.1  = 0.871     R.sub.2 = -6.682                  D = 1.020   n.sub.0 = 1.5g = 0.583 h.sub.4 = -0.657                  h.sub.6 = -0.832                              f = 1.0NA = 0.6  WD = 0.275Embodiment 41R.sub.1 = 0.816     R.sub.2 = -8.201                  D = 0.900   n.sub.0 = 1.5g = 0.583 h.sub.4 = -0.649                  h.sub.6 = -0.860                              f = 1.0NA = 0.6  WD = 0.329Embodiment 42R.sub.1 = 0.915     R.sub.2 = 7.404                  D = 1.070   n.sub.0 = 1.65g = 0.504 h.sub.4 = -0.579                  h.sub.6 = 0.942                              f = 1.0NA = 0.6  WD = 0.234______________________________________ 
    
     where R 1 , R 2  respectively represent the radii of curvatures of the lens surfaces, D represents the length of the lens, n 0  represents the refractive index on the optical axis of the lens, the parameter g represents the gradient of the refractive index distribution, h 4  and h 6  respectively represent the 4th- and 6th-order coefficients of the refractive index distribution, f represents the focal length of the lens, NA represents the numerical aperture at the side of the disk, and WD represents the distance between the lens and the disk. Both the coefficient g of the refractive index distribution and the value of refractive index are for the wave length λ=780 nm. 
     Among them, each of Embodiments 33, 34, 40 and 41 according to the present invention is a biconvex lens as shown in FIG. 6, and each of Embodiments 35 through 39 and 42 is a positive meniscus lens as shown in FIG. 7. 
     Each of Embodiments 43 through 51 according to the present invention shown in the following numerical data are the lenses satisfying the conditions (11) through (13). 
     
         ______________________________________Embodiment 43R.sub.1 = 1.075     R.sub.2 = 16.903                  D = 1.320   n.sub.0 = 1.65g = 0.521 h.sub.4 = -0.693                  h.sub.6 = -0.677                              f = 1.0NA = 0.5Embodiment 44R.sub.1 = 1.069     R.sub.2 = -15.880                  D = 1.368   n.sub.0 = 1.80g = 0.375 h.sub.4 = 1.058                  h.sub.6 = 21.635                              f = 1.0NA = 0.5Embodiment 45R.sub.1 = 1.035     R.sub.2 = 2.0750                  D = 1.248   n.sub.0 = 1.80g = 0.479 h.sub.4 = -0.954                  h.sub.6 = -2.156                              f = 1.0NA = 0.5Embodiment 46R.sub.1 = 1.022     R.sub.2 = 115.424                  D = 1.272   n.sub.0 = 1.65g = 0.500 h.sub.4 = -0.602                  h.sub.6  = 0.441                              f = 1.0NA = 0.6Embodiment 47R.sub.1 = 1.002     R.sub.2 = 4.042                  D = 1.200   n.sub.0 = 1.65g = 0.542 h.sub.4 = -0.844                  h.sub.6 = -1.800                              f = 1.0NA = 0.6Embodiment 48R.sub.1 = 1.039     R.sub.2 = 5.178                  D = 1.320   n.sub.0 = 1.80g = 0.417 h.sub.4 = -0.296                  h.sub.6 = 5.302                              f = 1.0NA = 0.6Embodiment 49R.sub.1 = 1.059     R.sub.2 = 2.671                  D = 1.320   n.sub.0 = 1.80g = 0.458 h.sub.4 = -0.854                  h.sub.6 = -1.367                              f = 1.0NA = 0.6Embodiment 50R.sub.1 = 1.060     R.sub.2 = 10.703                  D = 1.320   n.sub.0 = 1.65g = 0.521 h.sub.4 = -0.741                  h.sub. 6 = -0.911                              f = 1.0NA = 0.65Embodiment 51R.sub.1 = 1.074     R.sub.2 = 3.970                  D = 1.368   n.sub.0 = 1.80g = 0.438 h.sub.4 = -0.682                  h.sub.6 = -0.661                              f = 1.0NA = 0.65______________________________________ 
    
     where R 1 , R 2  respectively represent the radii of curvatures of the lens surfaces, D represents the length of the lens, n 0  represents the refractive index on the optical axis of the lens, the parameter g represents the gradient of the refractive index distribution, h 4  and h 6  respectively represent the 4th- and 6th-order coefficients of the refractive index distribution, f represents the focal length of the lens, and NA represents the numerical aperture at the side of the disk. Both the coefficient g of the refractive index distribution and the value of refractive index are for the wave length λ=780 nm. 
     Among them, each of Embodiments 43, 45 through 51 according to the present invention is a positive meniscus lens as shown in FIG. 9, and Embodiment 44 according to the present invention is a biconvex lens as shown in FIG. 8. 
     Embodiments 52 through 64 according to the present invention shown in the following the numerical data are the lenses satisfying the conditions (14) and (15). 
     
         ______________________________________Embodiment 52R.sub.1 = 1.697     R.sub.2 = -1.356                  D = 1.56    n.sub.0 = 1.5g = 0.583 h.sub.4 = 0.208                  h.sub.6 = 1.640                              f = 1.0NA = 0.5Embodiment 53R.sub.1 = 1.830     R.sub.2 = -1.518                  D = 1.68    n.sub.0 = 1.65g = 0.500 h.sub.4 = 0.530                  h.sub.6 = 3.966                              f = 1.0NA = 0.5Embodiment 54R.sub.1 = 2.195     R.sub.2 = -1.807                  D = 1.92    n.sub.0 = 1.65g = 0.542 h.sub.4 = 0.120                  h.sub.6 = 0.561                              f = 1.0NA = 0.5Embodiment 55R.sub.1 = 1.332     R.sub.2 = -2.166                  D = 1.56    n.sub.0 = 1.8g = 0.375 h.sub.4 = 1.755                  h.sub.6 =  26.713                              f = 1.0NA = 0.5Embodiment 56R.sub.1 = 1.657     R.sub.2 = -3.000                  D = 1.80    n.sub.0 = 1.8g = 0.458 h.sub.4 = -0.103                  h.sub.6 = 1.743                              f = 1.0NA = 0.5Embodiment 57R.sub.1 = 1.355     R.sub.2 = 1.906                  D = 1.56    n.sub.0 = 1.5g = 0.667 h.sub.4 = -0.415                  h.sub.6 = -0.316                              f = 1.0NA = 0.6Embodiment 58R.sub.1 = 1.500     R.sub.2 = -4.096                  D = 1.62    n.sub.0 = 1.5g = 0.625 h.sub.4 = -0.268                  h.sub.6 = 0.024                              f = 1.0NA = 0.6Embodiment 59R.sub.1 = 1.303     R.sub.2 = -374.044                  D = 1.56    n.sub.0 = 1.65g = 0.542 h.sub.4 = -0.570                  h.sub.6 = -0.430                              f = 1.0NA = 0.6Embodiment 60R.sub.1 = 1.771     R.sub.2 = -2.650                  D = 1.80    n.sub.0 = 1.65g = 0.542 h.sub.4 = -0.130                  h.sub.6 = 0.446                              f = 1.0NA = 0.6Embodiment 61R.sub.1 = 1.389     R.sub.2 = 2.792                  D = 1.68    n.sub.0 = 1.8g = 0.500 h.sub.4 = -0.747                  h.sub.6 = -1.277                              f = 1.0NA = 0.6Embodiment 62R.sub.1 = 1.322     R.sub.2 = -7.400                  D = 1.62    n.sub.0 = 1.8g = 0.438 h.sub.4 = -0.410                  h.sub.6 = 1.969                              f = 1.0NA = 0.6Embodiment 63R.sub.1 = 1.289     R.sub.2 = 47.625                  D = 1.56    n.sub.0 = 1.65g = 0.542 h.sub.4  = -0.601                  h.sub.6 = -0.530                              f = 1.0NA = 0.7Embodiment 64R.sub.1 = 1.233     R.sub.2 = 6.217                  D = 1.56    n.sub.0 = 1.8g = 0.458 h.sub.4 = -0.791                  h.sub.6 = -0.991                              f = 1.0NA = 0.7______________________________________ 
    
     where R 1 , R 2  respectively represent the radii of curvatures of the lens surfaces, D represents the length of the lens, n 0  represents the refractive index on the optical axis of the lens, the parameter g represents the gradient of the refractive index distribution, h 4  and h 6  respectively represent the 4th- and 6th-order coefficients of the refractive index distribution, f represents the focal length of the lens, and NA represents the numerical aperture at the side of the disk. Both the coefficient g of the refractive index distribution and the value of refractive index are for the wave length λ=780 nm. 
     Among them, each of Embodiments 52 through 56, 58 through 60 and 62 according to the present invention is a biconvex lens as shown in FIG. 10, and Embodiments 57, 61, 63 and 64 according to the present invention is a positive meniscus lens as shown in FIG. 11. 
     In Embodiments 1 through 21 of the above mentioned Embodiments according to the present invention, aberrations are corrected for the lens system involving the disk whose thickness and refractive index are 1.2 mm and 1.55 respectively, and in the aberration curves of the respective Embodiments shown as FIG. 12 through 32, the above mentioned disk is taken into account. 
     In Embodiments 22 through 64 according to the present invention, aberrations are also corrected for the lens system involving the disk whose thickness and refractive index are 0.288 mm and 1.55 respectively, and in the aberration curves of respective Embodiments, the above mentioned disk is also taken into account. 
     In Embodiments 1 through 64, all of higher-order coefficients than h 6  are regarded as zero. 
     As is mentioned above, in the GRIN single lens system according to the present invention, N.A. is large and various aberrations including off-axial aberration especially coma are corrected excellently.