We consider a single lens of thickness at the center. The distance from the object to the left surface is , and the distance from the right surface of the lens to the image is . The index of refraction of the lens is .

t

ta

tb

n

The goal is the construction of an aberration-free lens. This means the optical path lengths of all rays must be the same. Compared with the optical path length of a ray along the center line, any other ray has a larger geometric arc length. To achieve a constant optical path length, the longer geometric path length must be compensated by a smaller distance of the ray within the lens. Given the shape of the left surface of the lens in the form , the equality of all optical path lengths determines the exit points for every ray, and so the shape of the right lens surface.

z=z(r)

Using Snell's law, one can determine a parametrization of the right lens surface purely algebraically in terms of the shape of the left lens surface. The parametrization is in terms of the coordinate of the left surface, as the coordinate determines a unit entry point of the ray into the lens. No differential equation of the integral equation has to be solved. Such a construction is possible as long as the light rays don’t cross.

r

r

In the paper, the authors give Wolfram Language code to determine the right surface. Here is a slightly condensed form of their code:

In[1]:=

rightLensSurface[ra_,za_,n_,{ta_?Negative,t_,tb_?Positive}]:=Module{Φ,ri,zi,fi,z0,h0,h1,zb,rb,ζ=za[ra],d=[ra],α,β},α=Sqrt[+];β=(ra+(ζ-ta)d)(nα(1+));Φ=Sqrt1(1+)-;ri=β-dΦ;zi=d(ri+dΦ)+Φ;fi=ta-tb+α;h0=ζ+finzi-rarizi+(t+tb)-(ζ+tzi);h1=+2rarit+++-2t(tb-ζ)(zi-1);zb=h0+Sqrt[-2fin(rari+t+zi(t(zi-1)-tb+ζ))+h1-](1-);rb=ra+ri(zb-ζ)zi;{zb,rb}

′

za

2

ra

2

(ζ-ta)

2

d

2

d

2

β

2

ri

2

zi

2

n

2

ra

2

(tb-ζ)

2

t

2

ri

2

(zi-1)

2

zi

2

fi

2

ri

2

n

2

(razi+ri(t+tb-ζ))

2

n

Here is an example of the left lens surface being defined through a Bessel function:

In[2]:=

leftLensSurface[ra_]:=BesselJ[0,ra/2]-1

This is the resulting lens shape. The left lens surface is plotted as a blue curve, and the right one as a dark red curve:

lensCurves=ParametricPlot[{{leftLensSurface[ra],ra},rightLensSurface[ra,leftLensSurface,1.5,{-20,3,30}]},{ra,-6.15,6.15},PlotStyle{Blue,Darker[Red]},AxesLabel{"z","r"}]

Out[3]=

Gluing the two surfaces together gives the lens:

In[4]:=

lens=Polygon[Join[#[[1,1]],Reverse[#[[2,1]]]]&[Cases[lensCurves,_Line,∞]]];

We define the path of a light ray through the lens:

In[5]:=

rayPath[ra_]:=Line[{{-20,0},{leftLensSurface[ra],ra},rightLensSurface[ra,leftLensSurface,1.5,{-20,3,30}],{30+3,0}}]

Graphics[{LightGray,EdgeForm[Black],lens,Darker[Yellow],Thickness[0.001],Table[rayPath[ra],{ra,-6.,6,12/40}]},AxesTrue,ImageSize800]

Out[6]=

Here is an explicit check that all rays have the same optical path length:

In[7]:=

pathLength[Line[{p0_,pa_,pb_,p1_}],n_]:=EuclideanDistance[p0,pa]+nEuclideanDistance[pa,pb]+EuclideanDistance[pb,p1]

pathLength[#,1.5]&/@Table[rayPath[ra],{ra,-6.,6,12/20}]

Out[8]=

{54.5,54.5,54.5,54.5,54.5,54.5,54.5,54.5,54.5,54.5,54.5,54.5,54.5,54.5,54.5,54.5,54.5,54.5,54.5,54.5,54.5}

Combining the lens construction and the ray drawing into a single function AberrationFreeLensRayPlot allows us to conveniently plot many lens shapes:

In[9]:=

makeLens[za_,n_,{ta_?Negative,t_,tb_?Positive},R_]:=Module[{lensCurves},lensCurves=ParametricPlot[{{za[ra],ra},rightLensSurface[ra,za,n,{ta,t,tb}]},{ra,-R,R},Exclusions{},PlotStyleBlack,AxesLabel{"z","r"}];Polygon[Join[#[[1,1]],Reverse[#[[2,1]]]]&[Cases[lensCurves,_Line,∞]]]]

In[10]:=

rayPath[ra_,za_,n_,{ta_?Negative,t_,tb_?Positive}]:=Line[{{ta,0},{za[ra],ra},rightLensSurface[ra,za,n,{ta,t,tb}],{tb+t,0}}]

In[11]:=

AberrationFreeLensRayPlot[za_,n_,{ta_?Negative,t_,tb_?Positive},R_,rc_]:=Module[{lens=makeLens[za,n,{ta,t,tb},R]},Graphics[{LightGray,EdgeForm[Black],lens,Darker[Yellow],Thickness[0.001],cc=Cases[tab=Table[N@rayPath[ra,za,n,{ta,t,tb}],{ra,-0.9R,0.9R,1.8R/rc}],Line[{{_Real,_Real}..}]]},AxesTrue,ImageSize680]]

Here is the same functional shape of the lens surface, but with smaller object and image distances:

The next graphic uses a much larger index of refraction:

The following two plots use spherical left lens surfaces:

Here is a flat left lens surface:

Here is a conical left lens surface:

To get an intuitive feeling for the shape of the right lens surface as a function(al) of the left surface, we implement an interactive demonstration that defines the left lens surface as an interpolation of six control points: