fig1 illustrates the terminology used herein to define the nature of the index and dispersion gradients for grin lens elements shown are two grin lens elements arranged along a ( horizontal ) optical axis . the subscript fr refers to the front ( object ) side and the subscript r refers to the rear ( image ) side . the variables n and v refer respectively to the local index of refraction and the abbe number , v =[ n ( λ f )- n ( λ c )]/[ n ( λ d )- 1 ], of the grin material . thus , for the lens elements in fig1 the refractive index gradient is given as δn =( n fr - n r ) and the dispersion gradient is given as δv =( v fr - v r ) where the fr and r subscripts denote the planes perpendicular to the optical axis at the extreme positions of the lens surface . the subsequent figures included herein maintain this convention wherein the left side of the drawing is the object side and the right side is the image side , and the optical axis is a horizontal line . fig2 shows a cross section of a lens system having six lens elements e1 , . . . , e6 in accordance with a first embodiment of the present invention for use for instance as a high speed photographic objective lens . the ray tracings are generated by a computer program of a type well known in optical engineering . in this embodiment the grin lens elements are the first ( leftmost ) and the sixth ( rightmost ) lens elements respectively e1 and e6 . these grin lens elements e1 , e6 may be fabricated by the above - described method of u . s . pat . no . 4 , 929 , 065 . the aperture stop is depicted by the small vertical lines between the third e3 and fourth e4 lens elements . the lens prescription for this assembly is shown in table i . table i__________________________________________________________________________lens prescription for example 1 . rdy thi gla element__________________________________________________________________________obj : infinity infinity1 : 70 . 12595 5 . 800000 ` g1 ` e12 : 140 . 69079 0 . 1000003 : 41 . 59626 10 . 037292 803181 . 463799 e24 : - 77 . 56245 3 . 800000 603420 . 380299 e35 : 22 . 84637 6 . 496686sto : infinity 6 . 4208007 : - 21 . 24697 3 . 800000 784701 . 260768 e48 : - 765 . 92367 7 . 832550 803181 . 463799 e59 : - 32 . 08652 0 . 10000010 : 127 . 36280 7 . 400000 ` g10 ` e611 : - 50 . 07149 39 . 074360img : infinity - 0 . 048416wl 656 . 28 587 . 56 546 . 07 486 . 13 435 . 84 ` g1 ` 1 . 797984 1 . 803181 1 . 807300 1 . 815304 1 . 824957c1 - 0 . 3777e - 01 - 0 . 3802e - 01 - 0 . 3822e - 01 - 0 . 3859e - 01 0 . 3903e - 01 ` g10 ` 1 . 617272 1 . 620411 1 . 622870 1 . 627558 1 . 633122c1 0 . 2442e - 01 0 . 2470e - 01 0 . 2492e - 01 0 . 2537e - 01 0 . 2592e - 01infinite conjugates efl 55 . 0000 bfl 39 . 0744 ffl - 13 . 7478 fno 1 . 4230 img dis 39 . 0259 oal 51 . 7873paraxial image ht 21 . 6000 ang 21 . 4413entrance pupil dia 38 . 6506 thi 26 . 3965exit pupil dia 52 . 9536 thi - 36 . 2789__________________________________________________________________________ there is no requirement that the grin lens element be in the front and rear positions ; other locations and configurations of the grin lens elements were investigated and also found to provide substantial benefit . for instance in other embodiments , only one grin lens element was used and in others the grin lens elements were in locations other than shown in fig2 . the second through fifth lens elements e2 , . . . , e5 shown in fig2 in this example are constructed from conventional optical glasses . both lens doublets e2 and e3 , e4 and e5 proximate to the aperture stop are cemented to reduce manufacturing complexity . fig3 a and 3b depict the expected optical performance of the lens assembly shown in fig2 . fig3 a shows the spherical aberration , astigmatic , and distortion correction achieved by the lens assembly . fig3 b shows the polychromatic transverse ray aberration plot of the lens assembly at various positions in the field - of - view . the lens configuration of fig2 exhibits good monochromatic and polychromatic aberration correction . the first grin lens element e1 is used primarily to correct spherical aberration and coma while the second grin lens element e6 is used primarily to correct astigmatism and chromatic aberrations . since the front grin lens element e1 has a negligible dispersion gradient , δv ( λ 2 )≅ 0 , it does not provide significant chromatic aberration correction but , is effective in balancing monochromatic aberration since the index gradient is fairly large , δn ( λ 2 )≅- 0 . 221 . the grin material is defined by a base index of refraction and a linear slope coefficient : z = axial distance from the plane perpendicular to the optical axis at the extreme of the front surface , where n 0 ( λ ) is the index of refraction at wavelength , λ , at the plane z = 0 . in this case , the grin material for element e1 is defined in table ii at five different wavelengths λ 1 - λ 5 , the c -, d -, e -, f , and g - lines respectively . table ii______________________________________description of front grin lens element , example 1 . wavelength ( nm ) n . sub . 0 ( λ ) c . sub . 1 ( λ ) ______________________________________λ . sub . 1 656 . 28 1 . 797984 - 0 . 03777λ . sub . 2 587 . 56 1 . 803181 - 0 . 03802λ . sub . 3 546 . 07 1 . 807300 - 0 . 03822λ . sub . 4 486 . 13 1 . 815304 - 0 . 03859λ . sub . 5 435 . 84 1 . 824957 - 0 . 03903______________________________________ since the c 1 coefficient of this grin material is negative , the index of refraction decreases as one moves from the front to the back of the lens element . the rear grin lens element e6 has a dispersion gradient , δv ( λ 2 )≅- 13 . 9 , and provides chromatic correction in addition to monochromatic aberration correction made possible with an index gradient , δn ( λ 2 )≅+ 0 . 200 . table iii represents the data used to model this grin material of the rear element e6 . table iii______________________________________description of rear grin lens element , example 1 . wavelength ( nm ) n . sub . 0 ( λ ) c . sub . 1 ( λ ) ______________________________________λ . sub . 1 656 . 28 1 . 617272 + 0 . 02442λ . sub . 2 587 . 56 1 . 620411 + 0 . 02470λ . sub . 3 546 . 07 1 . 622870 + 0 . 02492λ . sub . 4 486 . 13 1 . 627558 + 0 . 02537λ . sub . 5 435 . 84 1 . 633122 + 0 . 02592______________________________________ since the c 1 coefficient in of this grin material is positive , the index of refraction increases as one moves from the front to the back of the lens element e6 . a second lens system in accordance with the present invention is shown in fig4 again including six lens elements e &# 39 ; 1 , . . . , e &# 39 ; 6 where again the front element e &# 39 ; 1 and rear element e &# 39 ; 6 are grin lens elements . this lens system is intended to improve the optical performance over that of the first lens system of fig2 by further correcting higher order field dependent aberrations and by exploiting the aberration correction capabilities of the grin lens further . this was achieved by altering the lens system of fig2 by separating the two lens elements of the first cemented doublet ( the second and third lens elements e &# 39 ; 2 and e &# 39 ; 3 ) and thus allowing an &# 34 ; air lens &# 34 ; to form between lens elements e &# 39 ; 2 and e &# 39 ; 3 . this results in improved optical performance , especially near the edge of the field . there is , however , a slight increase in secondary color . detail of the optical performance of the lens system of fig4 is shown respectively in fig5 a and 5b and the corresponding lens prescription is shown in table iv . table iv__________________________________________________________________________lens prescription for example 2 . rdy thi gla element__________________________________________________________________________obj : infinity infinity1 : 107 . 97905 6 . 700000 ` g1 ` e &# 39 ; 12 : 0185 . 91914 0 . 1000003 : 30 . 72433 8 . 661343 840421 . 433470 e &# 39 ; 24 : 50 . 87642 1 . 7443225 : 152 . 21160 2 . 000000 688930 . 310790 e &# 39 ; 36 : 23 . 18947 6 . 451361sto : infinity 5 . 9578228 : - 24 . 47914 2 . 800000 728250 . 283410 e &# 39 ; 49 : 85 . 23343 8 . 346506 840421 . 433470 e &# 39 ; 510 : - 37 . 41160 0 . 10000011 : 96 . 12875 7 . 800000 ` g11 ` e &# 39 ; 612 : - 53 . 69734 38 . 097298img : infinity - 0 . 048416wl 656 . 28 587 . 56 546 . 07 486 . 13 435 . 84 ` g1 ` 1 . 791419 1 . 796681 1 . 800858 1 . 808978 1 . 818801c1 - 0 . 3439e - 01 - 0 . 3475e - 01 - 0 . 3504e - 01 - 0 . 3562e - 01 0 . 3634e - 01 ` g11 ` 1 . 514322 1 . 516800 1 . 518723 1 . 522384 1 . 526703c1 0 . 4106e - 01 0 . 4149e - 01 0 . 4183e - 01 0 . 4252e - 01 0 . 4336e - 01infinite conjugates efl 54 . 9998 bfl 38 . 0973 ffl - 16 . 2817 fno 1 . 4111 img dis 38 . 0973 oal 49 . 8614paraxial image ht 21 . 6000 ang 21 . 4414entrance pupil dia 38 . 9761 thi 29 . 1148exit pupil dia 47 . 2213 thi - 28 . 5374__________________________________________________________________________ in order to further correct higher order optical aberrations present in the lens system of fig2 the first cemented doublet e2 , e3 of fig2 was separated in the embodiment of fig4 and the resulting air lens between elements e &# 39 ; 2 and e &# 39 ; 3 in fig4 provided . in addition , the configuration was altered compared to fig2 such that both the front and rear grin lens elements e &# 39 ; 1 and e &# 39 ; 6 have dispersion gradients in addition to index gradients . this allows increased freedom to use the grin lens properties to balance chromatic aberrations while using optical power distribution and / or lens bending to balance the monochromatic aberrations . the front grin lens element e &# 39 ; 1 has an index gradient , δn ( λ 2 )≅- 0 . 233 , and a dispersion gradient , δv ( λ 2 )≅+ 15 . 3 , and as such provides both monochromatic aberration and chromatic aberration correction for the entire lens assembly . table v represents the data used to model the grin material of the front element e &# 34 ; 1 . table v______________________________________description of front grin lens element , example 2 . wavelength ( nm ) n . sub . 0 ( λ ) c . sub . 1 ( λ ) ______________________________________λ . sub . 1 656 . 28 1 . 791419 - 0 . 03439λ . sub . 2 587 . 56 1 . 796681 - 0 . 03475λ . sub . 3 546 . 07 1 . 800858 - 0 . 03504λ . sub . 4 486 . 13 1 . 808978 - 0 . 03526λ . sub . 5 435 . 84 1 . 818801 - 0 . 03634______________________________________ since the c 1 coefficient of this grin material is negative , the index of refraction decreases as one moves from the front to the back of the lens element . the rear grin lens element e &# 39 ; 6 has an index gradient , δn ( λ 2 )≅+ 0 . 324 , and a dispersion gradient , δv ( λ 2 )≅- 20 . 8 , and as such provides both monochromatic aberration and chromatic aberration correction for the entire lens assembly . table vi represents the data used to model the grin material of the rear element e &# 39 ; 6 . table vi______________________________________description of rear grin lens element , example 2 . wavelength ( nm ) n . sub . 0 ( λ ) c . sub . 1 ( λ ) ______________________________________λ . sub . 1 656 . 28 1 . 514322 - 0 . 04106λ . sub . 2 587 . 56 1 . 516800 - 0 . 04149λ . sub . 3 546 . 07 1 . 518723 - 0 . 04183λ . sub . 4 486 . 13 1 . 522384 - 0 . 04252λ . sub . 5 435 . 84 1 . 526703 - 0 . 04336______________________________________ since the c 1 coefficient of this grin material is positive , the index of refraction increases as one moves from the front to the back of the lens element e &# 39 ; 6 . in this lens system both the front e &# 39 ; 1 and the rear e &# 39 ; 6 grin lens elements have significant index and dispersion gradients and as such each contributes to the overall correction of monochromatic as well as chromatic aberration correction . the lens systems embodiments shown here are illustrative and are not limiting ; further modifications in the location of the grin lens elements and of the nature and location of the non - grin lens are possible in accordance with the present invention . also , other qualities of the various grin and non - grin lens elements may be altered as will be well understood by one of ordinary skill in the art . it is especially found that grin lens elements are substitutable ( with suitable design changes ) for aspherical lens elements . more generally , it has been found that the ability of a lens element to corrector balance aberrations in an optical system is also related to the optical power such a lens element has compared to the optical power of the entire system . with that in mind , a ratio of the optical power of the grin lens element to that of the total lens system is defined herein with reference to fig1 . this ratio provides a simple measure of the overall effect an individual grin lens element has on its ability to affect the aberration of the total optical system . therefore it has been found that in accordance with the invention and as illustrated in the above examples , the following properties are present : a six lens element photographic objective lens system which utilizes two positive optical power linear axial grin lens elements in the following locations : one grin lens element before an aperture stop where n fr & gt ; n r , and one grin lens element after the aperture stop where n fr & lt ; n r , per fig1 . each of the front and rear grin lens elements also satisfies at least one of the following conditions : δn = n fr - n r and δv = v fr - v r as in fig1 . as to relative optical power , the grin lens element in the front group satisfies : thus , by combining the above conditions for certain embodiments of the invention , the above description is illustrative and not limiting , further modifications will be apparent to one skilled in the art in light of this disclosure and are intended to fall within the scope of the appended claims .