Abstract:
A compact gradient index rod lens that can be manufactured without decreasing the amount of incident light. The gradient index rod lens includes a lens body radially distributing refractive indexes. The lens body has a cross sectional outline formed by removing at least part of a peripheral portion of an original lens body.

Description:
BACKGROUND OF THE INVENTION 
     The present invention relates to a gradient index rod lens, a method for manufacturing a gradient index rod lens, and a lens array. 
     A gradient index rod lens is manufactured by performing a treatment, such as ion exchange, on a cylindrical piece of glass. This distributes refractive indexes from the central portion of the lens to the peripheral portion of the lens. The refractive indexes at the peripheral portion of the lens normally are not included in the intended gradient index rod range. Thus, the peripheral portion of the lens cannot be used. For example, referring to FIG. 27, a gradient index rod lens  11  has an effective portion  12 . The effective portion  12  is located in the center of the gradient index rod lens  11  and has a diameter, which is denoted by De. Aberrations are tolerated In the effective portion  12 . A peripheral portion  13  is defined around the effective portion  12 . The refractive indexes distributed in the peripheral portion  13  are not included in the intended range. Accordingly, the diameter Do of the gradient index rod lens  11  is determined by adding a value obtained by multiplying the width of the peripheral portion by two to the effective diameter De of the effective portion  12 . Since the refractive indexes distributed in the peripheral portion  13  are not included in the tolerable range, the focal point of the light that passes through the peripheral portion  13  differs from that of the light that passes through the effective portion  12 . This produces a large aberration in the lens as shown by FIG.  28 . 
     The conventional gradient index rod lens has a diameter that is significantly greater than the effective diameter of the effective portion  12 , which is the portion actually functioning as a lens. Thus, when a plurality of gradient index rod lens are arranged to form a lens array  21 , as shown in FIG. 29, the lenses increases the size of the lens array  21  and the pitch between effective portions  12 . This lowers resolution. In the example shown in FIG. 29, the gradient index rod lenses  11  are arranged in V-shaped grooves of a substrate  22 . To reduce the size of the lens array  21 , the diameter Do of the lens may be decreased. However, this would decrease the area of the effective portion and decrease the amount of the light that enters the effective portion  12 . 
     A planar micro-lens array has been proposed to decrease the size of a lens array. In one type of lens array, the gradient index rod is such that the refractive index differs at different depths in a substrate. In another type of lens array, the surface of a lens array is etched to form pits, and resins hating different refractive indexes are filled in the pits. However, satisfactory lens characteristics cannot be obtained when such lens arrays are used to connect optical fibers with optical devices. 
     SUMMARY OF THE INVENTION 
     It is an object of the present invention to provide a smaller gradient index rod lens without decreasing the amount of light that enters its effective portion. 
     To achieve the above object, the present invention provides a gradient index rod lens including a lens body radially distributing refractive indexes. The lens body has a cross sectional outline formed by removing at least part of a peripheral portion of a cylindrical original lens body. 
     A further perspective of the present invention is a method for manufacturing a gradient index rod lens. The method includes preparing a cylindrical original lens body and forming a lens body having a predetermined cross sectional outline by removing at least part of a peripheral portion of the original lens body. 
     A further perspective of the present invention is a lens array including at least a row of a plurality of gradient index rod lenses. Each of the gradient index rod lenses includes a lens body radially distributing refractive indexes. The lens body has a cross sectional outline formed by removing at least part of a peripheral portion of a cylindrical original lens body. 
     Other aspects and advantages of the present invention will become apparent from the following description, taken in conjunction with the accompanying drawings, illustrating by way of example the principles of the invention. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The invention, together with objects and advantages thereof, may beat be understood by reference to the following description of the presently preferred embodiments together with the accompanying drawings in which: 
     FIG. 1 is a perspective view showing a gradient index rod lens according to a first embodiment of the present invention; 
     FIG. 2 is an explanatory diagram illustrating the aberration of the gradient index rod lens of FIG. 1; 
     FIG. 3 is a perspective view showing a further gradient index rod lens according to the first embodiment of the present invention; 
     FIG. 4 is a perspective view showing a lens array including the gradient index rod lens of FIG. 1; 
     FIG. 5 is a perspective view showing a lens array including the gradient index rod lens of FIG. 3; 
     FIG. 6 is a schematic cross-sectional view illustrating a procedure for machining a gradient index rod lens in a second embodiment of the present invention; 
     FIG. 7 is a front view showing the gradient index rod lens of the second embodiment; 
     FIG. 8 is a front view showing a lens array including the gradient index rod lens of FIG. 7; 
     FIG. 9 is a perspective view showing a two-dimensional array including the gradient index rod lens of FIG. 7; 
     FIG. 10 is a front view showing a gradient index rod lens according to a third embodiment of the present invention; 
     FIG. 11 is a front view showing a lens array including the gradient index rod lens of FIG. 10; 
     FIG. 12 is a perspective view showing a two-stage lens array including the gradient index rod lens of FIG. 7; 
     FIG. 13 is a perspective view showing a two-dimensional array including the gradient index rod lens of FIG. 10; 
     FIG. 14 is a front view showing a gradient index rod lens according to a fourth embodiment of the present invention; 
     FIG. 15 is a front view showing a lens array including the gradient index rod lens of FIG. 14; 
     FIG. 16 is a perspective view of a two-dimensional array including the gradient index rod lens of FIG. 14; 
     FIG. 17 is a front view showing a gradient index rod lens according to a fifth embodiment of the present invention; 
     FIG. 18 is a front view showing a lens array including the gradient index rod lens of FIG. 17; 
     FIG. 19 is a front view showing a further gradient index rod lens according to the fifth embodiment of the present invention; 
     FIG. 20 is a front view showing a lens array including the gradient index rod lens of FIG. 19; 
     FIG. 21 is a front view of a gradient index rod lens according to a first modification of the fifth embodiment; 
     FIG. 22 is a front view showing a lens array including the gradient index rod lens of FIG. 21; 
     FIG. 23 is a front view of a gradient index rod lens according to a second modification of the fifth embodiment; 
     FIG. 24 is a front view showing a lens array including the gradient index rod lens of FIG. 23; 
     FIG. 25 is a front view showing a gradient index rod lens according to the third embodiment of the present inventions; 
     FIG. 26 is a front view showing a two-stage lens array including the gradient index rod lens of FIG. 1; 
     FIG. 27 is a perspective view showing a prior art gradient index rod lens; 
     FIG. 28 is an explanatory diagram illustrating the aberration of the prior art gradient index rod lens; and 
     FIG. 29 is a front view showing a lens array including the prior art gradient index rod lens. 
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     In the drawings, like numerals are used for like elements throughout. 
     First Embodiment 
     A gradient index rod lens  14  and a lens array  16  according to a first embodiment of the present invention will now be described with reference to FIGS. 1 to  5 . 
     In FIG. 1, the broken lines show a gradient index rod lens  11  (original lens), which is manufactured through a normally performed process. The gradient index rod lens  11  undergoes mechanical or chemical processing to partially remove its peripheral portion  13 . The partial removal of the peripheral portion  13  exposes an underlying peripheral portion  13   a  of the lens  14 . When the diameter of the lens  14  prior to the removal of the peripheral portions  13   a  is Do (refer to FIG.  27 ), the diameter D of the lens  14  of FIG. 1 subsequent to the removal of the peripheral portions  13   a  is smaller than Do. It is preferred that the peripheral portion  13  be partially removed so that the ratio between the effective diameter De of an effective portion  12  (indicated by diagonal lines) and the lens diameter D be 0.3≦De/D≦1. In one example, the peripheral portions of a gradient index rod lens  11  satisfying the equation of De/Do=0.4 are removed to form a gradient index rod lens  14  satisfying the equation of De/D=0.9. FIG. 2 illustrates the aberration when only the effective portion  12  of the gradient index rod lens  14  is irradiated with light. As apparent from FIG. 2, the aberration is small. The peripheral portion may be removed by performing mechanical processing, or cylindrical grinding. If chemical processing is performed, the gradient index rod lens  11  is dipped into a hydrofluoric acid solution to partially remove the peripheral portions. The two ends of the lens  11  are then mechanically ground to form the gradient index rod lens  14 . Further, all of the peripheral portions  13  may be removed (De/D=1) to form the lens  14 . In this case, light enters only the effective portion  12 . Thus, the aberration is similar to that shown in FIG.  2 . 
     As shown in FIG. 3, all of the peripheral portion and part of the effective portion  12  may be removed to form a lens  15 . In this case, the effective diameter Df of the lens subsequent to the removal satisfies Df&lt;De. Thus, the entire gradient index rod lens  15  is the effective portion. The aberration of the lens is similar to that illustrated in FIG. 2 
     FIG. 4 is a schematic view showing a lens array  16  including a plurality of the gradient index rod lenses  14 , from which the peripheral portions  13  have partially been removed. The lens array  16  is formed by embedding a plurality of the gradient index rod lenses  14  in a synthetic resin substrate. The effective diameter De of the effective portion  12  in each lens  14  is the same as that of the effective diameter in a conventional lens. However, the diameter D of each lens  14  is smaller than the diameter Do of the conventional lens. This decreases the pitch between the lenses  14 , which, in turn, increases the resolution of the lens array  16 . 
     FIG. 5 is a schematic view showing a lens array  16  including a plurality of the gradient index rod lenses  15 , from which all of the peripheral portions  13  and part of the effective portions  12  have partially been removed. In this case, the diameter of each lens  15  is smaller (Df&lt;De) than that of the lens  14 . Thus, the lens array  16  is more compact than that of FIG.  4  and has a higher resolution. 
     The gradient index rod lenses of the first embodiment have the advantages described below. 
     (1) In the gradient index rod lens  14 , a predetermined amount of part of the peripheral portion  13  is removed from the lens  14 , and the effective portion  12  remains intact in the lens  14 . Thus, the lens  14  is made more compact without touching the effective portion  12 . Further, the area occupied by the effective portion  12  in the gradient index rod lens  14  is greater than that of a conventional lens having the same lens diameter. Thus, light enters a wider effective portion. In other words, the amount of light entering the effective portion of the lens is increased. 
     (2) In the gradient index rod lens  15 , part (outer portion) of the effective portion  12  is removed in addition to the peripheral portion  13 . Thus, the effective portion occupies the entire lens  15 , and the lens  15  is made more compact. Further, light enters an effective portion that is wider than that of a conventional lens having the same lens diameter. In other words, the amount of light entering the effective portion of the lens is increased. 
     (3) Mechanical processing, such as grinding, or chemical processing, such as etching, is performed to remove the peripheral portion  13 . Thus, the amount of removed material may be accurately controlled by measuring the removed amount during the processing. In other words, the lenses  14 ,  15  may be accurately formed. 
     (4) The removal of the peripheral portions  13  forms the compact lenses  14 ,  15 . Thus, the pitch of the lenses  14 ,  15  is small. As a result, the lens array  16  has a high resolution and a large capacity. 
     (5) In the lens array  16  that includes the gradient index rod lenses  14 ,  15 , which have the large effective portions  12 , the amount of light entering the effective portions  12  is large. 
     Second Embodiment 
     A gradient index rod lens  27  and a lens array  28  according to a second embodiment of the present invention will now be described with reference to FIGS. 6 to  9 . 
     Referring to FIG. 7, the gradient index rod lens  27  of the second embodiment has the form of a rectangular block. Further, the gradient index rod lens  27  is formed by machining the gradient index rod lens  11  to remove part of the peripheral portion  13  from the gradient index rod lens  11 . The cross-section of the rectangular block is such that it corresponds to squares ranging from one circumscribing the effective portion  12  with each side having a length of De to one inscribing the circumference of the lens  11  prior to machining with each side having a length of Dg. 
     The manufacturing of the lens  27  will now be described. As shown in FIG. 6, a plurality of the cylindrical gradient index rod lenses (rod lenses)  11  are arranged on a table  24 . Wax  23  is applied to adhere and fix the lenses  11  to the table  24 . The peripheral portions  13  of the lenses  11  is ground starting from a plane F that is parallel to the table  24  and tangential to the peripheral surfaces of the lenses  11 . A first surface, or one of the four surfaces, of each lens  27  is formed when a predetermined amount d 1  is ground. Referring to FIG. 7, the predetermined amount d 1  is minimal when the length of each of the four sides of the lens  27  is Dg and maximal when the length of each of the four sides is De. After the grinding is completed, the wax  23  is melted to remove the lenses  11  from the table  24 . The lenses  11  are then flipped over and adhered to the table  24  so that the ground surfaces contact the table  24 . The predetermined amount d 1  is then ground from the peripheral portions  13  on the other side of the ground surfaces in the same manner to form a second surface of each lens  27 . Subsequently, the lenses  11  are rearranged on and fixed to the table  24  so that the two ground surfaces are perpendicular to the table  24 . The predetermined amount d 1  is ground to form a third surface of each lens  27 . Then, the lenses  11  are flipped over and fixed to the table  24  to grind the predetermined amount d 1  from the remaining peripheral portions  13  and form a fourth surface of each lens  27 . This completes the formation of the rectangular block-like gradient index rod lenses  27 . 
     FIG. 8 is a schematic view showing the lens array  25  in which a plurality of the gradient index rod lenses  27  are arranged. The lens array  25  includes a substrate  26  and a plurality of the rectangular block-like gradient index rod lenses  27 , which are arranged on the substrate  26 . Since the lenses  27  have flat surfaces, the V-groove substrate  22  used in the prior art lens array  21 , which is shown in FIG. 29, is not necessary. Further, since the peripheral portion  13  of each lens  27  is removed, the pitch between the lenses  27  is small. Thus, the lens array  25  has a resolution that is higher than that of the prior art lens array  21  shown in FIG.  29 . 
     FIG. 9 is a schematic view showing a lens array  28  in which a plurality of the gradient index rod lenses  27  are arranged in a two-dimensional manner. Rows of the gradient index rod lenses  27  are superimposed on the substrate  26 . When the lenses  27  are arranged in a two-dimensional manner, the pitch between the adjacent lenses  27  is relatively small. Thus, the two-dimensional lens array  28  has a high resolution. 
     In addition to advantages (1), (3), (4), and (5) of the first embodiment, the gradient index rod lens  27  of the second embodiment has the advantages described below. 
     (6) The lens  27  has the form of a rectangular block. This decreases the pitch between the lenses  27  and increases the resolution in one-dimensional and two-dimensional arrays. 
     (7) The lens  27  has a flat bottom surface (peripheral surface). Thus the lens array  28  is formed just by placing the lens  27  on the flat substrate  26 . In other words, the pitch of the lenses  27  is accurately set without using an expensive V-groove substrate. 
     Third Embodiment 
     A gradient index rod lens  30  and lens arrays  29 ,  31 ,  32  according to a third embodiment of the present invention will now be described with reference to FIGS. 10 to  13 . 
     Referring to FIG. 10, the gradient index rod lens  30  of the third embodiment has the form of a triangular block. Further, the gradient index rod lens  30  is formed by machining the gradient index rod lens  11  to remove part of the peripheral portion  13  from the cylindrical gradient index rod lens  11 . The cross-section of the triangular block is such that it corresponds to a triangle ranging from one circumscribing the effective portion  12  with each side having a length of Te to one inscribing the circumference of the lens  11  prior to machining with each side having a length of Tg. 
     Since the lens  30  is manufactured in the same manner as the rectangular block-like lens  27  of the second embodiment, only the differing points will be described. The peripheral portions  13  of a plurality of the gradient index rod lenses  11 , which are arranged on a table, are ground parallel to the table  24 . A first surface, or one of the three surfaces, of each lens  30  is formed when a predetermined amount d 1  is ground. The predetermined amount d 1  is minimal when the length of each of the three sides of the lens  30  is Tg and maximal when the length of each of the three sides is Te. After the grinding is completed, the lenses  11  are removed from the table  24 . The lenses  11  are then adhered to the table  24  in a state in which the first surfaces are inclined by 60° relative to the table  24 . The predetermined amount d 1  is then ground from the peripheral portions  13  to form a second surface of each lens  30 . Subsequently, the lenses  11  are rearranged on and fixed to the table  24  so that the first and second surfaces are inclined by 60° relative to the table  24 . The predetermined amount d 1  is ground to form a third surface of each lens  30 . This completes the formation of the triangular block-like gradient index rod lenses  30 . 
     FIG. 11 is a schematic view showing the lens array  29 , which includes a plurality of the gradient index rod lenses  30 . The triangular block-like gradient index rod lenses  30  are arranged close to each other on a substrate  26 . The lenses  30  have a flat bottom surface. Thus, the V-groove substrate used by the lens array  21  of FIG. 29 is unnecessary. Further, the two-dimensional lens array  31 , which has a two-stage structure as shown in FIG. 12, is formed by arranging a plurality of the refractive distribution index lenses  30  between adjacent lenses  30 , which have been arranged on the substrate  26 . 
     FIG. 13 is a schematic view showing the lens array  32 , which includes three or more stages (in this case, four stages) of the triangular block-like gradient index rod lenses  30 . In other words, a plurality of the triangular block-like gradient index rod lenses  30  are superimposed on the substrate  26 . 
     In addition to advantages (1), (3), (4), (5), and (7) of the first and second embodiments, the gradient index rod lens  30  and the lens arrays  29 ,  31 ,  32  of the third embodiment has the advantage described below. 
     (8) The pitch of the triangular block-like lenses  30  is smaller than that of the cylindrical lenses  11 . Thus, the one-dimensional lens array  29  and the two-dimensional lens arrays  31 ,  32  have a high resolution. 
     Fourth Embodiment 
     A gradient index rod lens  34  and lens arrays  33 ,  35  according to a fourth embodiment of the present invention will now be described with reference to FIGS. 14 to  16 . 
     Referring to FIG. 14, the gradient index rod lens  34  of the fourth embodiment has the form of a hexagonal block. Further, the gradient index rod lens  34  is formed by machining the gradient index rod lens  11  to remove part of the peripheral portion  13  from the cylindrical gradient index rod lens  11 . The cross-section of the hexagonal block is such that it corresponds to a hexagon ranging from one circumscribing the effective portion  12  with each side having a length of He to one inscribing the circumference of the lens  11  prior to machining with each side having a length of Hg. 
     Since the lens  34  is manufactured in the same manner as the rectangular block-like lens  27  of the second embodiment, only the differing points will be described. The peripheral portions  13  of a plurality of the gradient index rod lenses  11  are arranged on a table  24  and ground parallel to the table  24 . A first surface, or one of the six surfaces, of each lens  34  is formed when a predetermined amount d 1  is ground. The predetermined amount d 1  is minimal when the length of each of the six sides of the lens  34  is Hg and maximal when the length of each of the six sides is He. After the grinding is completed, the lenses  11  are removed from the table  24 . Subsequently, the lenses  11  are ground to sequentially form second to sixth surfaces. Thus, the lenses  11  undergo grinding for a total of six times. This completes the formation of the hexagonal block-like gradient index rod lenses  34 . 
     FIG. 15 is a schematic view showing the lens array, which includes a plurality of the gradient index rod lenses  34 . The hexagonal block-like gradient index rod lenses  34  are arranged on a substrate  26  in a state contacting side surfaces of the adjacent lenses  34 . Since the lenses  34  have flat side surfaces, the side surfaces are easily connected with each other. FIG. 16 is a schematic view showing the lens array  35 , which includes a plurality of the gradient index rod lenses  34  accumulated so as to have a dense structure. 
     In addition to advantages (1), (3), (4), (5), and (7) of the first and second embodiments, the gradient index rod lens  34  and the lent arrays  33 ,  35  of the fourth embodiment have the advantage described below. 
     (9) The hexagonal lens  34  is optimal for forming the two-dimensional lens array  35 , which has a dense structure. 
     Fifth Embodiment 
     Gradient index rod lens  36 ,  39  and lens arrays  38 ,  40  according to a fifth embodiment of the present invention will now be described with reference to FIGS. 17 to  19 . 
     As shown in FIG. 17, the gradient index rod lens  36  of the fifth embodiment has two parallel flat side surfaces  37   a . The lens  36  is formed by machining the peripheral portion  13  to remove two side portions  31  , which are located on opposite sides of the effective portion  12 , from a lens  11 . Each of the side surfaces  37   a  is formed by grinding the side portions  37  until reaching the effective portion  12 . The grinding is performed in the same manner as in the second embodiment. 
     FIG. 18 is a schematic view showing the lens array  38 , which includes a plurality of the lenses  36 . The lens array  38  includes a substrate  22 . V-shaped grooves  22   a  extend along the surface of the substrate  22 . The gradient index rod lenses  36  are each arranged in one of the V-shaped grooves  22   a . The V-shaped grooves  22   a  are formed by performing anisotropic etching or by dicing the substrate  22  with a diamond blade saw. The distance between adjacent V-shaped grooves  22   a  is determined by the distance between the two side surfaces  38   a  of each gradient index rod lens  36 . Two arcuate surfaces extend between the two side surfaces  37   a  in each gradient index rod lens  36 . Each lens  36  is arranged in the corresponding V-shaped groove  22   a  with parts of one of its arched surfaces contacting parts of the V-shaped groove  22   a . Further, the adjacent lenses  36  are in contact with each other. Due to the closely arranged lenses  36 , the lens array  38  has a high resolution. 
     As shown in FIG. 19, the gradient index rod lens  39  has one flat side surface  37   b . In this case, as shown in FIG. 20, the lens  39  is arranged on the substrate  22  so that the side surface  37   b  is perpendicular to the substrate  22 . This decreases the pitch between the adjacent lenses  39 . Thus, the lens array  40  has a high resolution. 
     In addition to advantages (1), (3), (4), and (5) of the first embodiment, the gradient index rod lens  36  and the lens arrays  38  of the fifth embodiment have the advantage described below. 
     ( 10 ) The lens  39  is arranged so that its arcuate surface contacts the V-shaped groove  22   a  and its side surface  37   b  is perpendicular to the substrate  22 . Thus, the lens array  40  has a high resolution. 
     A first modification and a second modification of the lens  39  of the fifth embodiment will now be described with reference to FIGS. 21 to  24 . 
     (First Modification) 
     To form a lens array that employs V-shaped grooves  22   a  of a substrate  22 , a lens  41  having a cornered portion, as shown in FIG. 21, is fitted in each of the V-shaped grooves  22   a . The cornered portion of the lens  41  is defined by two side surfaces  41   a , which are formed by grinding a cylindrical lens. The angle θ between the two side surfaces  41   a  of the cornered portion is 90°. As shown in FIG. 22, the cornered portion is fitted in the corresponding V-shaped groove  22   a , the surfaces of which intersect at 90°. A plurality of the lenses  41  is arranged on the substrate  22  in the same manner to form the lens array  42 , 
     (Second Modification) 
     Referring to FIG. 23, a gradient index rod lens  43  has a cornered portion defined by two side surfaces  43   a , which are formed by grinding a cylindrical lens until reaching the effective portion  12 . The angle φ between the two side surfaces  43   a  of the cornered portion is 60°. As shown in FIG. 24, the lens  43  is arranged on a substrate  22  so that the cornered portion is fitted in a corresponding V-shaped groove  22   a , the surfaces of which intersect at 60°. A plurality of the lenses  43  is arranged on the substrate  22  in the same manner to form the lens array  44 . 
     FIG. 25 shows the triangular block-like gradient index rod lens  30  of the third embodiment. The angle between the two sides of the triangular block-like gradient index rod lens  30  is 60°. A plurality of the lens  30  may be fitted in the 60° V-shaped grooves  22   a  to form a lens array on the substrate  22 . 
     In addition to advantages (1), (3), (4), and (5) of the first embodiment, the first and second modifications have the advantage described below. 
     (11) The gradient index rod lenses  30 ,  41 ,  43  are cornered with an angle corresponding to the V-shaped grooves  22   a . Thus, the lenses  30 ,  41 ,  43  are easily and accurately attached to the substrate  22 . 
     It should be apparent to those skilled in the art that the present invention may be embodied in many other specific forms without departing from the spirit or scope of the invention. Particularly, it should be understood that the present invention may be embodied in the following forms. 
     The cylindrical gradient index rod lenses  14  or  15  of the first embodiment may be arranged in the V-shaped grooves  22   a , as shown in FIG.  26 . Further lenses  14 ,  15  may be superimposed on the lenses  14 ,  15 , which have been arranged in the V-shaped grooves  22   a , to form a lens array  45  having a two-dimensional structure. This facilitates the arrangement of the cylindrical gradient index rod lens  14 ,  15 . 
     The rectangular block-like gradient index rod lenses  27  of the second embodiment may be arranged in the V-shaped grooves of a substrate to form a lens array. In this case, the angle between the adjacent side surfaces of each lens is 90°. It is thus preferred that the angle of the V-shaped grooves be 90°. 
     The hexagonal block-like gradient index rod lenses  34  of the fourth embodiment may be arranged in the V-shaped grooves of a substrate to form a lens array. In this case, the angle between the adjacent side surfaces of each lens is 120°. It is thus preferred that the angle of the V-shaped grooves be 120°. 
     The gradient index rod lenses  36  or  39  of the fifth embodiment may be arranged on a flat substrate so that their flat surfaces  37   a ,  37   b  contact the surface of the substrate to form a lens array. A two-dimensional lens array may also be formed by a arranging a plurality of the lenses  36  on a substrate in this manner. In this case, the height of the gradient index rod lens  36  is relatively low. Thus, the two-dimensional lens array is relatively low. 
     The present examples and embodiments are to be considered as illustrative and not restrictive, and the invention is not to be limited to the details given herein, but may be modified within the scope and equivalence of the appended claims.