[ 0019 ] fig1 ( prior art ) shows a conventional optical imaging system using a contact lens over an eye . object 15 is imaged through contact lens 25 through the cornea 26 , the iris 27 , lens 28 , and the vitreous humor 29 onto the retina 30 . such a system creates a sharp , in - focus image at the retina 30 only if object 15 is located at or very close to the in - focus object plane . some accommodation is provided by the lens 28 . however , lens hardens 28 with age and loses its ability to refocus . if the distance from the back principal plane of lens 28 to retina 30 is d i , and the focal length of contact lens 25 is f , the distance from the front principal plane of lens 28 to object 15 , d o must be chosen such that : 1 d o + 1 d i - 1 f = 0 ( eq .  1 ) in order for the image at retina 30 to be in adequate focus . the depth of field of an optical system is the distance that the object 15 can move away from the in - focus distance and still have the image be in focus . for a simple system like fig1 the depth of focus is very small , unless the light is bright and the iris is stopped down . prior attempts to solve this problem have used contact lenses and optical implants that have multiple ( usually two ) foci . one focus is correct for objects at infinity , and one is correct for objects at a close distance . this means that two images of an object at one of those locations are formed , one in focus and one out of focus . [ 0022 ] fig2 shows this effect when imaging a point at infinity . rays 41 form a point image 50 at one of the foci of the combined system formed by contact lens 25 , cornea 26 , and lens 28 . the second focus of the system forms a blurred image 52 . when the point object is at a reading distance , the previously blurred image 52 is in focus , and the image 50 becomes blurred . at other distances , neither image is in focus , and the degree of misfocus changes with object location . p ( x , y )= e j ( αx 3 + βy 3 + γx 2 y + δxy 2 ) , x 2 + y 2 ≦ 1 . ( eq . 2 ) choice of the constants , α , β , γ and δ allow phase functions that are rectangularly separable ( with γ = δ = 0 ) to systems whose modulation transfer functions ( mtf ) are circularly symmetric ( α = β = α 0 , γ = δ =− 3α 0 ). for simplicity we will use the symmetric rectangularly separable form , which is given by : p ( x , y )= e ja ( x 3 + y 3 ) , x 2 + y 2 ≦ 1 , ( eq . 3 ) where α is a parameter used to adjust the depth of field increase . since this form is rectangularly separable , for most analyses its one - dimensional component can be considered : p ( x , y )= e jαx 3 , x 2 ≦ 1 . ( eq . 4 ) as the absolute value of α increases , the depth of field increases . the image contrast before post - processing also decreases as α increases . this is because as α increases , the mtf slumps down . [ 0028 ] fig3 shows the effect of the edf - coding phase element on the rays that pass through the eye . rays 41 , which come from a point at infinity and pass through contact lens 65 , cornea 26 , and lens 28 , do not form a useful focus in the eye . phase coding might be applied by variations in the thickness of lens 65 ( exaggerated here for clarity ). an expanded view of the ray pattern near the retina 30 is shown in expanded view 70 where the retina is at plane 72 . this is unlike the ray pattern for an eye with a normal contact lens and unlike the ray pattern for an eye with two - foci lens of fig2 . as the object point moves in from infinity , the ray pattern in the region of the retina 30 , shown expanded in 70 , moves to the left , but the cross section of the ray pattern that falls on the retina 30 does not change appreciably . accordingly , the substantially same ray pattern will fall on the retina without regard to object location . when the object is not a point , the object distribution is convolved with the unchanging ray distribution ( e . g ., the point spread function ). the brain can de - convolve the resulting coded image because the point spread function of the eye , modified with the edf - coding optics , does not change appreciably . this is in contrast to the changes that normally occur in the point spread function when there is misfocus . for large enough α , the mtf of a system using a cubic phase modulation ( pm ) mask can be approximated by :  h   ( u , ψ ) ≈ π 3   α   u  , u ≠ 0 , and ( eq .  5 ) where u is the spatial frequency in the x direction and ψ is the degree of misfocus . thus , the cubic - pm mask is an example of a mask which modifies the optical system to have a near - constant mtf over a range of object distances . the particular range for which the mtf does not substantially vary depends on α . this range , and thus the depth of field , increases with α . however , the amount that the depth of field can be increased is practically limited by the fact that contrast decreases as α increases . for the human eye , moderate increases in depth of field are typically needed . [ 0032 ] fig4 shows an edf intraocular implant imaging system in accordance with one embodiment where the edf coding optical shape is on the implant lens 75 . the phase coding results in misfocus as shown in expanded view 70 of fig3 . [ 0033 ] fig5 shows an edf modified natural eye imaging system in accordance with one embodiment where the edf coding optical shape is on the cornea 85 . the cornea 85 can be modified using laser surgery . for example , phase coding is applied through laser - produced variations in the thickness of cornea 85 ( thickness of cornea 85 exaggerated here for clarity ). the phase coding results in misfocus as shown in expanded view 70 of fig3 .