fig1 shows the arrangement of lens elements in collimator lens systems in accordance with first to seventh embodiments of the present invention . as shown in fig1 each collimator lens system comprises first to third lens elements l 1 to l 3 in three groups arranged in this order from the parallel light bundle side ( from the left in fig1 ). the first lens element l 1 is a positive double - convex lens , the second lens element l 2 is a negative lens concave to the parallel light bundle side and the third lens element l 3 is a positive lens convex to the parallel light bundle side . the following formulae ( 1 ) to ( 5 ) are satisfied wherein f represents the overall focal length of the collimator lens system , f i represents the focal length of i - th lens element , i standing for 1 , 2 or 3 , r i represents the radius of curvature of i - th refracting surface as numbered from the side of the parallel light bundle , i standing for 1 , 2 , 3 , 4 , 5 or 6 ( being a positive value when the refracting surface is convex to the side of the parallel light bundle and a negative value when the refracting surface is concave to the side of the parallel light bundle ) and n i represents the refractive index of i - th lens , i standing for 1 , 2 or 3 . when formulae ( 1 ) to ( 5 ) are satisfied , the numerical aperture can be relatively large and at the same time the sine condition can be excellent . accordingly coma does not greatly deteriorate even if the optical axis is shifted relative to the light source and setting of the optical axis to the light source ( alignment adjustment ) is facilitated . further when the formulae ( 1 ) and ( 3 ) are satisfied , the back focal length can be within a practical range , whereby positioning of the collimator lens system relative to the light source is facilitated . when the formula ( 3 ) is satisfied , the radius of curvature r 5 of the parallel light bundle side face of the third lens l 3 may not be so small and accordingly lens processing is facilitated . further when the formulae ( 2 ), ( 3 ) and ( 5 ) are satisfied , spherical aberration can be well corrected . in fig1 g denotes a window of the laser diode ( perpendicular to the optical axis x ) and t denotes the thickness of the window g . the radii of curvature r of the refracting surfaces , the axial surface separations d ( the central thicknesses of the lens elements or the air separations ) and the refractive indexes n of the lens elements in the collimator lens system in accordance with a first embodiment of the present invention are as shown in the following table 1 . in table 1 , the radii of curvature r , the axial surface separations d and the refractive indexes n are designated in order from the parallel light bundle side at r 1 to r 6 , d 1 to d 5 and n 1 to n 3 , respectively . the radii of curvature r , the axial surface separations d and the thickness t of the window g of the laser diode are in the values standardized on the basis of the overall focal length f (= 1 ) of the collimator lens system . ( the same for all the embodiments ) table 1______________________________________r d n______________________________________r . sub . 1 = 0 . 9264 d . sub . 1 = 0 . 3343 n . sub . 1 = 1 . 61333r . sub . 2 = - 0 . 9264 d . sub . 2 = 0 . 1067r . sub . 3 = - 0 . 5615 d . sub . 3 = 0 . 0571 n . sub . 2 = 1 . 76593r . sub . 4 = 2 . 2784 d . sub . 4 = 0 . 0143r . sub . 5 = 0 . 6020 d . sub . 5 = 0 . 0239 n . sub . 3 = 1 . 61333r . sub . 6 = - 1 . 3981______________________________________f = 1 . 0 , na = 0 . 3 , t = 0 . 0119 , λ = 780 nm ( 1 ) f . sub . 1 / f . sub . 3 = 1 . 128 ( 2 ) r . sub . 2 ( n . sub . 2 - 1 )/{ r . sub . 3 ( n . sub . 1 - 1 )} = 2 . 060 ( 3 ) r . sub . 5 /{ f ( n . sub . 3 - 1 )} = 0 . 982 ( 4 ) - r . sub . 1 / r . sub . 2 = 1 . 000 ( 5 ) - f . sub . 2 / f = 0 . 583______________________________________ fig2 a shows the spherical aberration ( the solid line ) and the sine condition ( the broken line ) at 780 nm of the collimator lens system of the first embodiment as the collimator lens system is regarded as a focusing lens system for focusing a light bundle from the parallel light bundle side . as can be understood from fig2 a , the collimator lens system of this embodiment is excellent in both the spherical aberration and the sine condition though it is very simple in structure and the numeric aperture na is relatively large ( na = 0 . 3 ). thus the collimator lens system of this embodiment is easy to manufacture . the radii of curvature r of the refracting surfaces , the axial surface separations d ( the central thicknesses of the lens elements or the air separations ) and the refractive indexes n of the lens elements in the collimator lens system in accordance with a second embodiment of the present invention are as shown in the following table 2 . in table 2 , the radii of curvature r , the axial surface separations d and the refractive indexes n are designated in order from the parallel light bundle side at r 1 to r 6 , d 1 to d 5 and n 1 to n 3 , respectively . table 2______________________________________r d n______________________________________r . sub . 1 = 1 . 4895 d . sub . 1 = 0 . 1997 n . sub . 1 = 1 . 61333r . sub . 2 = - 0 . 8658 d . sub . 2 = 0 . 0767r . sub . 3 = - 0 . 6523 d . sub . 3 = 0 . 0830 n . sub . 2 = 1 . 76593r . sub . 4 = ∞ d . sub . 4 = 0 . 0160r . sub . 5 = 0 . 6775 d . sub . 5 = 0 . 0198 n . sub . 3 = 1 . 61333r . sub . 6 = - 3 . 0441______________________________________f = 1 . 0 , na = 0 . 3 , t = 0 . 0083 , λ = 780 nm ( 1 ) f . sub . 1 / f . sub . 3 = 1 . 000 ( 2 ) r . sub . 2 ( n . sub . 2 - 1 )/{ r . sub . 3 ( n . sub . 1 - 1 )} = 1 . 658 ( 3 ) r . sub . 5 /{ f ( n . sub . 3 - 1 )} = 1 . 105 ( 4 ) - r . sub . 1 / r . sub . 2 = 1 . 720 ( 5 ) - f . sub . 2 / f = 0 . 852______________________________________ in this embodiment , the light source side face of the second lens element l 2 is flat ( r 4 =∞). fig2 b shows the spherical aberration ( the solid line ) and the sine condition ( the broken line ) at 780 nm of the collimator lens system of the second embodiment as the collimator lens system is regarded as a focusing lens system for focusing a light bundle from the parallel light bundle side . as can be understood from fig2 b , the collimator lens system of this embodiment is excellent in both the spherical aberration and the sine condition though it is very simple in structure and the numeric aperture na is relatively large ( na = 0 . 3 ). thus the collimator lens system of this embodiment is easy to manufacture . the radii of curvature r of the refracting surfaces , the axial surface separations d ( the central thicknesses of the lens elements or the air separations ) and the refractive indexes n of the lens elements in the collimator lens system in accordance with a third embodiment of the present invention are as shown in the following table 3 . in table 3 , the radii of curvature r , the axial surface separations d and the refractive indexes n are designated in order from the parallel light bundle side at r 1 to r 6 , d 1 to d 5 and n 1 to n 3 , respectively . table 3______________________________________r d n______________________________________r . sub . 1 = 1 . 3556 d . sub . 1 = 0 . 2304 n . sub . 1 = 1 . 61657r . sub . 2 = - 1 . 3556 d . sub . 2 = 0 . 1312r . sub . 3 = - 0 . 9100 d . sub . 3 = 0 . 0792 n . sub . 2 = 1 . 79413r . sub . 4 = 7 . 0417 d . sub . 4 = 0 . 0084r . sub . 5 = 0 . 5782 d . sub . 5 = 0 . 2648 n . sub . 3 = 1 . 61657r . sub . 6 = 3 . 2039______________________________________f = 1 . 0 , na = 0 . 33 , t = 0 . 0066 , λ = 675 nm ( 1 ) f . sub . 1 / f . sub . 3 = 1 . 031 ( 2 ) r . sub . 2 ( n . sub . 2 - 1 )/{ r . sub . 3 ( n . sub . 1 - 1 )} = 1 . 919 ( 3 ) r . sub . 5 /{ f ( n . sub . 3 - 1 )} = 0 . 938 ( 4 ) - r . sub . 1 / r . sub . 2 = 1 . 000 ( 5 ) - f . sub . 2 / f = 1 . 010______________________________________ in this embodiment , the light source side face of the third lens element l 3 ( r 6 ) is concave ( convex to the parallel light bundle side ). fig2 c shows the spherical aberration ( the solid line ) and the sine condition ( the broken line ) at 675 nm of the collimator lens system of the third embodiment as the collimator lens system is regarded as a focusing lens system for focusing a light bundle from the parallel light bundle side . as can be understood from fig2 c , the collimator lens system of this embodiment is excellent in both the spherical aberration and the sine condition though it is very simple in structure and the numeric aperture na is relatively large ( na = 0 . 33 ). thus the collimator lens system of this embodiment is easy to manufacture . the radii of curvature r of the refracting surfaces , the axial surface separations d ( the central thicknesses of the lens elements or the air separations ) and the refractive indexes n of the lens elements in the collimator lens system in accordance with a fourth embodiment of the present invention are as shown in the following table 4 . in table 4 , the radii of curvature r , the axial surface separations d and the refractive indexes n are designated in order from the parallel light bundle side at r 1 to r 6 , d 1 to d 5 and n 1 to n 3 , respectively . table 4______________________________________r d n______________________________________r . sub . 1 = 3 . 4887 d . sub . 1 = 0 . 5089 n . sub . 1 = 1 . 63079r . sub . 2 = - 1 . 7664 d . sub . 2 = 0 . 1111r . sub . 3 = - 0 . 9662 d . sub . 3 = 0 . 3356 n . sub . 2 = 1 . 76593r . sub . 4 = - 1 . 7778 d . sub . 4 = 0 . 0578r . sub . 5 = 0 . 9560 d . sub . 5 = 0 . 7111 n . sub . 3 = 1 . 76593r . sub . 6 = - 6 . 8167______________________________________f = 1 . 0 , na = 0 . 35 , t = 0 . 0556 , λ = 780 nm ( 1 ) f . sub . 1 / f . sub . 3 = 1 . 694 ( 2 ) r . sub . 2 ( n . sub . 2 - 1 )/{ r . sub . 3 ( n . sub . 1 - 1 )} = 2 . 220 ( 3 ) r . sub . 5 /{ f ( n . sub . 3 - 1 )} = 1 . 248 ( 4 ) - r . sub . 1 / r . sub . 2 = 1 . 975 ( 5 ) - f . sub . 2 / f = 3 . 367______________________________________ in this embodiment , the light source side face of the second lens element l 2 ( r 4 ) is convex ( concave to the parallel light bundle side ). fig2 d shows the spherical aberration ( the solid line ) and the sine condition ( the broken line ) at 780 nm of the collimator lens system of the fourth embodiment as the collimator lens system is regarded as a focusing lens system for focusing a light bundle from the parallel light bundle side . as can be understood from fig2 d , the collimator lens system of this embodiment is excellent in both the spherical aberration and the sine condition though it is very simple in structure and the numeric aperture na is relatively large ( na = 0 . 35 ). thus the collimator lens system of this embodiment is easy to manufacture . the radii of curvature r of the refracting surfaces , the axial surface separations d ( the central thicknesses of the lens elements or the air separations ) and the refractive indexes n of the lens elements in the collimator lens system in accordance with a fifth embodiment of the present invention are as shown in the following table 5 . in table 5 , the radii of curvature r , the axial surface separations d and the refractive indexes n are designated in order from the parallel light bundle side at r 1 to r 6 , d 1 to d 5 and n 1 to n 3 , respectively . table 5______________________________________r d n______________________________________r . sub . 1 = 1 . 7749 d . sub . 1 = 0 . 3322 n . sub . 1 = 1 . 6133r . sub . 2 = - 1 . 7749 d . sub . 2 = 0 . 2444r . sub . 3 = - 0 . 9417 d . sub . 3 = 0 . 2867 n . sub . 2 = 1 . 76593r . sub . 4 = - 2 . 3947 d . sub . 4 = 0 . 1111r . sub . 5 = 0 . 5556 d . sub . 5 = 0 . 4100 n . sub . 3 = 1 . 61333r . sub . 6 = 3 . 1037______________________________________f = 1 . 0 , na = 0 . 41 , t = 0 . 0333 , λ = 780 nm ( 1 ) f . sub . 1 / f . sub . 3 = 1 . 443 ( 2 ) r . sub . 2 ( n . sub . 2 - 1 )/{ r . sub . 3 ( n . sub . 1 - 1 )} = 2 . 354 ( 3 ) r . sub . 5 /{ f ( n . sub . 3 - 1 )} = 0 . 906 ( 4 ) - r . sub . 1 / r . sub . 2 = 1 . 000 ( 5 ) - f . sub . 2 / f = 2 . 216______________________________________ in this embodiment , the light source side face of the second lens element l 2 ( r 4 ) is convex ( concave to the parallel light bundle side ) and the light source side face the third lens element l 3 ( r 6 ) is concave ( convex to the parallel light bundle side ). fig2 e shows the spherical aberration ( the solid line ) and the sine condition ( the broken line ) at 780 nm of the collimator lens system of the fifth embodiment as the collimator lens system is regarded as a focusing lens system for focusing a light bundle from the parallel light bundle side . as can be understood from fig2 e , the collimator lens system of this embodiment is excellent in both the spherical aberration and the sine condition though it is very simple in structure and the numeric aperture na is relatively large ( na = 0 . 41 ). thus the collimator lens system of this embodiment is easy to manufacture . the radii of curvature r of the refracting surfaces , the axial surface separations d ( the central thicknesses of the lens elements or the air separations ) and the refractive indexes n of the lens elements in the collimator lens system in accordance with a sixth embodiment of the present invention are as shown in the following table 6 . in table 6 , the radii of curvature r , the axial surface separations d and the refractive indexes n are designated in order from the parallel light bundle side at r 1 to r 6 , d 1 to d 5 and n 1 to n 3 , respectively . table 6______________________________________r d n______________________________________r . sub . 1 = 2 . 3114 d . sub . 1 = 0 . 4327 n . sub . 1 = 1 . 63079r . sub . 2 = - 2 . 3114 d . sub . 2 = 0 . 0654r . sub . 3 = - 1 . 0497 d . sub . 3 = 0 . 2500 n . sub . 2 = 1 . 63079r . sub . 4 = - 1 . 7939 d . sub . 4 = 0 . 2423r . sub . 5 = 0 . 7442 d . sub . 5 = 0 . 4327 n . sub . 3 = 1 . 63079r . sub . 6 = ∞ ______________________________________f = 1 . 0 , na = 0 . 3 , t = 0 . 024 , λ = 780 nm ( 1 ) f . sub . 1 / f . sub . 3 = 1 . 611 ( 2 ) r . sub . 2 ( n . sub . 2 - 1 )/{ r . sub . 3 ( n . sub . 1 - 1 )} = 2 . 202 ( 3 ) r . sub . 5 /{ f ( n . sub . 3 - 1 )} = 1 . 180 ( 4 ) - r . sub . 1 / r . sub . 2 = 1 . 000 ( 5 ) - f . sub . 2 / f = 4 . 610______________________________________ in this embodiment , the light source side face of the second lens element l 2 ( r 4 ) is convex ( concave to the parallel light bundle side ) and the light source side face the third lens element l 3 ( r 6 ) is flat . fig2 f shows the spherical aberration ( the solid line ) and the sine condition ( the broken line ) at 780 nm of the collimator lens system of the sixth embodiment as the collimator lens system is regarded as a focusing lens system for focusing a light bundle from the parallel light bundle side . as can be understood from fig2 f , the collimator lens system of this embodiment is excellent in both the spherical aberration and the sine condition though it is very simple in structure and the numeric aperture na is relatively large ( na = 0 . 3 ). thus the collimator lens system of this embodiment is easy to manufacture . the radii of curvature r of the refracting surfaces , the axial surface separations d ( the central thicknesses of the lens elements or the air separations ) and the refractive indexes n of the lens elements in the collimator lens system in accordance with a seventh embodiment of the present invention are as shown in the following table 7 . in table 7 , the radii of curvature r , the axial surface separations d and the refractive indexes n are designated in order from the parallel light bundle side at r 1 to r 6 , d 1 to d 5 and n 1 to n 3 , respectively . table 7______________________________________r d n______________________________________r . sub . 1 = 1 . 6280 d . sub . 1 = 0 . 3311 n . sub . 1 = 1 . 58254r . sub . 2 = - 1 . 6280 d . sub . 2 = 0 . 2444r . sub . 3 = - 0 . 8756 d . sub . 3 = 0 . 2867 n . sub . 2 = 1 . 76593r . sub . 4 = - 2 . 2112 d . sub . 4 = 0 . 1111r . sub . 5 = 0 . 5553 d . sub . 5 = 0 . 4096 n . sub . 3 = 1 . 58254r . sub . 6 = 6 . 0624______________________________________f = 1 . 0 , na = 0 . 42 , t = 0 . 0185 , λ = 780 nm ( 1 ) f . sub . 1 / f . sub . 3 = 1 . 421 ( 2 ) r . sub . 2 ( n . sub . 2 - 1 )/{ r . sub . 3 ( n . sub . 1 - 1 )} = 2 . 445 ( 3 ) r . sub . 5 /{ f ( n . sub . 3 - 1 )} = 0 . 953 ( 4 ) - r . sub . 1 / r . sub . 2 = 1 . 000 ( 5 ) - f . sub . 2 / f = 2 . 087______________________________________ in this embodiment , the light source side face of the second lens element l 2 ( r 4 ) is convex ( concave to the parallel light bundle side ) and the light source side face the third lens element l 3 ( r 6 ) is concave ( convex to the parallel light bundle side ). fig2 g shows the spherical aberration ( the solid line ) and the sine condition ( the broken line ) at 780 nm of the collimator lens system of the seventh embodiment as the collimator lens system is regarded as a focusing lens system for focusing a light bundle from the parallel light bundle side . as can be understood from fig2 g , the collimator lens system of this embodiment is excellent in both the spherical aberration and the sine condition though it is very simple in structure and the numeric aperture na is relatively large ( na = 0 . 42 ). thus the collimator lens system of this embodiment is easy to manufacture .