Ray Tracing 

SLAC and Stanford University Software Notices: 

Introduction 

• This package is designed to enable students to study simple optical systems beyond the usual thin lens approximation.  As structured it is limited to the study of lens systems where the lenses have a cylindrical  symmetry, so that it suffices to trace all rays in a single plane.  The source code can be easily adapted to the more general case of lenses with astigmatism.  In order to make it easy to develop more general code based on this package I expose the internal routines used by the workhouse function RayTrace .

 

• To use the basic package one only needs to understand the functions MakeLens , RayTrace and display .  All of the other functions are only needed to design extensions to the basic package.

 

• Each command in the RayTracing package can be accessed by using either the long form or the short form of the command name in the command calling sequence.

 

• As the underlying implementation of the <Package_Name> package is a module, it is also possible to use the form <Package_Name>:-command to access a command from the package. For more information, see Module Members .

 

 

Starting 

>	with(RayTracing);

 

"running setup" 

 

 

To start a raytracing worksheet begin with the command with(RayTracing) .  The restart command is at the top of this page so I can 

clear all variables and start fresh without shutting down Maple. 

 

The first thing you have to do when raytracing is define the left and right lens surfaces.  This is done by specifying two functions, the left hand lens surface is specified by the function and the right hand side by .  Note that the variable y runs along the vertical axis and the function gives the set-back of the lens from the origin of the horizontal axis.  Once we have them we can plot the lens by plotting two curves as shown below. 

 

Examples 

>	lens1:=y->evalf(13-sqrt(13^2-y^2)); lens2:=y->evalf(sqrt(13^2-y^2)-13+2*lens1(3)+.3); lens:=plot({[lens1(y),y,y=-3..3],[lens2(y),y,y=-3..3]}): lens;

 

 

 

 

>	MakeLens(lens1,lens2):

 

 

Later we want to deal with non-concentric lenses.  This is accomplished below.  Eventually we will define to be and then we can retrace 

everything to see the effects.
 

>	lens3:=x->evalf(sqrt(13^2-x^2)-13+2*lens1(3)+.4); shifted_lens:=plot({[lens1(x),x,x=-3..3],[lens3(x),x,x=-3..3]}): shifted_lens; lens4:=x->evalf(sqrt(13^2-x^2)-13+2*lens1(3)+.5);

 

 

 

 

>

_l1(3);

 

 

>

_l2(0);

 

 

>	focal_length(3);

 

 

 

The command RayTrace(p0,y_initial,index_of_refraction) to trace a single ray.  The first argument to the function is the 2-d vector specifying the 

starting point of the ray.  The syntax <x0,y0> is Maple's shorthand way of specifying a vector.  The next argument is the height at which you want 

the first ray to hit the first surface of the lens.  The last argument is the index of refraction for Joel's material. 

>	display(RayTrace(<-10,1.5>,1.5,1.4735),RayTrace(<-10,1>,1,1.4735),RayTrace(<-10,.5>,.5,1.4735), RayTrace(<-10,-.5>,-.5,1.4735), RayTrace(<-10,-1>,-1,1.4735),RayTrace(<-10,-1.5>,-1.5,1.4735),lens);

 

 

 

The plot below is the same but I have scaled it, and panned around to display the point of intersection and use Maple's ability to 

get coordinates off the plot to calculate the point-spread for a point source at infinity. 

 

>	display(RayTrace(<-10,1.5>,1.5,1.4735),RayTrace(<-10,1>,1,1.4735),RayTrace(<-10,.5>,.5,1.4735), RayTrace(<-10,-.5>,-.5,1.4735), RayTrace(<-10,-1>,-1,1.4735),RayTrace(<-10,-1.5>,-1.5,1.4735),lens);

 

 

 

The width of the spot at its minimum is 0.015mm, or 15 microns.  This is the best resolution you will get with a lens masked to a 3mm diameter.  It is about 

twice as bad if we go out to 5mm diameter masking.

One can also raytrace the imaging of point sources at a finite distance from the lens on axis. 

 

>	p0:=<-40,0>; display(RayTrace(p0,1.5,1.4735),RayTrace(p0,1,1.4735),RayTrace(p0,.5,1.4735), RayTrace(p0,-.5,1.4735), RayTrace(p0,-1,1.4735),RayTrace(p0,-1.5,1.4735),lens);

 

 

 

 

The point spread for the source at a finite distance is 0.017mm or 17microns.  Not much worse.  Some more traces for on axis points 

are shown below.  Note that the image moves toward the focal point as the source moves further away. 

 

>

 

>	p0:=<-100,0>;display(RayTrace(p0,1.5,1.4735),RayTrace(p0,1,1.4735),RayTrace(p0,.5,1.4735), RayTrace(p0,-.5,1.4735), RayTrace(p0,-1,1.4735),RayTrace(p0,-1.5,1.4735),lens);

 

 

 

>	p0:=<-200,0>;display(RayTrace(p0,1.5,1.4735),RayTrace(p0,1,1.4735),RayTrace(p0,.5,1.4735), RayTrace(p0,-.5,1.4735), RayTrace(p0,-1,1.4735),RayTrace(p0,-1.5,1.4735),lens);

 

 

 

 

Now we get to have some fun.  What is the point spread for the image of an off-axis point?  Scaling and measuring we get 0.018 mm.  Worse but not bad. 

Note the image is off axis.  From this we get the magnification (or demagnification of a distant object by the lens.)
 

>	p0:=<-100,.5>;display(RayTrace(p0,1.5,1.4735),RayTrace(p0,1,1.4735),RayTrace(p0,.5,1.4735), RayTrace(p0,-.5,1.4735), RayTrace(p0,-1,1.4735),RayTrace(p0,-1.5,1.4735),lens);

 

 

 

 

Some more off axis plots.  As we get more off axis the point spread gets only marginally worse 0.02mm.  This is not the cause of our fuzziness. 

 

>	p0:=<-100,2>;display(RayTrace(p0,1.5,1.4735),RayTrace(p0,1,1.4735),RayTrace(p0,.5,1.4735), RayTrace(p0,-.5,1.4735), RayTrace(p0,-1,1.4735),RayTrace(p0,-1.5,1.4735),lens);

 

 

 

>	p0:=<-100,2>;display(RayTrace(p0,1.5,1.4735),RayTrace(p0,1,1.4735),RayTrace(p0,.5,1.4735), RayTrace(p0,-.5,1.4735), RayTrace(p0,-1,1.4735),RayTrace(p0,-1.5,1.4735),lens);

 

 

 

>	p0:=<-100,2>;display(RayTrace(p0,1.5,1.4735),RayTrace(p0,1,1.4735),RayTrace(p0,.5,1.4735), RayTrace(p0,-.5,1.4735), RayTrace(p0,-1,1.4735),RayTrace(p0,-1.5,1.4735),lens);

 

 

 

 

Lets now redefine so as to do some non-concentric problems. 

 

>	lens2:=x->evalf(sqrt(13^2-(x+.05)^2)-13+2*lens1(3)+.02);

 

 

>	MakeLens(lens1,lens2): lens:=plot({[lens1(x),x,x=-3..3],[lens2(x),x,x=-3..3]}): lens;

 

 

 

Let's begin with a point source at infinity. 

 

>	display(RayTrace(<-10,1.5>,1.5,1.4735),RayTrace(<-10,1>,1,1.4735),RayTrace(<-10,.5>,.5,1.4735), RayTrace(<-10,-.5>,-.5,1.4735), RayTrace(<-10,-1>,-1,1.4735),RayTrace(<-10,-1.5>,-1.5,1.4735),lens);

 

 

>

 

 

 

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