Optical System Matrix and Conjugate Planes

Calculating the optical system matrix

Matlab code

function  F = sysmatrix(cv,th,rn)
%SYSMATRIX
%
%	F = SYSMATRIX(CV,TH,RN)
%
%    Returns the 2 x 2 paraxial system matrix given
%
%     CV   curvatures
%     TH   distance to following surface
%     RN   refractive index of following medium
%
%     first surface is a dummy to give index of incident medium
%
	m = length(th);
	if (nargin==2)
	   phi = cv;
	   td = th;
	else
	   phi(1) = 0;
	   td(1) = th(1)/rn(1);
	   for (i=2:m)
	      phi(i) = (rn(i)-rn(i-1))*cv(i);
	      td(i) = th(i)/rn(i);
	   end
	end
	F = eye(2,2);
	for (i=1:m)
	  F = F * [ 1 -phi(i); 0 1 ] * [ 1 0; td(i) 1];
	end

Example

>> th=[0 3.0486 5.4981 1.0162 0.6215 4.6129  3.0486  0 ];
>> rn=[1  1.713   1      1.689   1      1       1.713  1];
>> cv=[0  0.0458621 0.0039265 -0.0301655 0.0454845 0 0.0104227 -0.0406946];
>> 
>> f = sysmatrix(cv,th,rn)

f =

    0.8285   -0.0200
   17.6707    0.7806

Finding Conjugate Planes

Matlab code

function d = conjugates(m,f)
% CONJUGATES(M,F)
%
%   calculates object/image conjugates given the
%   system matrix and magnification
%
%        D = CONJUGATES(M,F)
%
%          where D(1) = object distance (object to system)
%                D(2) = image distance (system to image)
%                   M = magnification
%                   F = 2 x 2 optical system matrix

%  Author: John Loomis,  1 Oct 1997

    d = [ (1/m - f(2,2))/f(1,2)  (m - f(1,1))/f(1,2) ];

Example

>> bfd = conjugates(0,f)

Warning: Divide by zero.

bfd = -Inf   41.4387

>> pp = conjugates(1,f)

pp =   -10.9727   -8.5803

>> efl = bfd - pp

efl =  -Inf   50.0190

>> hiatus = sum(th) + sum(pp)

hiatus = -1.7070

Drawing an Optical System

Matlab code

function  [z,y] = drawlens(ap,th,cv,zstart)
%
% DRAWLENS(AP,TH,CV)
%
%

if (nargin<4)
   zstart = 0;
end

u = linspace(-ap,ap,60);

z = [ sag(u,cv(1)) th+sag(-u,cv(2)) sag(-ap,cv(1)) ] + zstart;
y = [ u  -u -ap ];
if (nargout==0) 
  plot(z,y)
end

Example

drawlens(9,th(2),cv(2:3));               
hold
drawlens(7.5,th(4),cv(4:5),sum(th(1:3)));
drawlens(9,th(7),cv(7:8),sum(th(1:6)));
axis([-1 19 -10 10]);
axis equal
axis square
hold off

Maintained by John Loomis, last updated 2 Oct 1997