Defocus and distance

Calculate is the displacement of the image plane required to produce a given defocus for lenses with different optical power.

Remember the simple lens maker's formula

  1/do + 1/di = 1/f

In the code below

baseD is 1/f (base lens power)
deltaD is the change in power.
1/do is zero (do is infinite) and thus the term never appears

We solve for the change in the image distance, di, to achieve a change in power. We call the change in image distance the displacement.

No matter the base lens power, to achieve a specific change in power we displace the image plane a specific fraction of the focal distance.

Imageval Consulting, LLC, 2018

Contents

Set up the base focal length

baseD  = 50:100:350;    % Diopters (power) of the base lens
deltaD = 1:15;          % Difference in diopters from the base power

ieNewGraphWin;
set(gca,'yscale','log');
hold on
for ii=1:length(baseD)
    % Lens maker's formula
    displacement = (1/baseD(ii)) - ( 1 ./ (baseD(ii) + deltaD));
    semilogy(deltaD,displacement);
end
xlabel('\Delta Diopters')
ylabel('Displacement (m)');
grid on

C = cell(size(baseD));
for ii=1:numel(baseD)
    C{ii} = sprintf('%d D',baseD(ii));
end
legend(C,'Location','northwest')

For any baseD, the \Delta diopter is a fraction of the focal length

% Equal ratios of diopters (deltaD/baseD) correspond to equal
% ratio of displacement/focal length
baseD  = 50:50:300;    % Diopters of focal length
deltaD = baseD/10;     % Choose a fraction.  Doesn't matter.
displacement = (1 ./ baseD) - ( 1 ./ (baseD + deltaD));

% The image plane displacement divided by the focal length is a
% constant for all of the different lens powers.
fprintf('Ratio of displacement to focal length %f\n',displacement.*baseD);

Ratio of displacement to focal length 0.090909
Ratio of displacement to focal length 0.090909
Ratio of displacement to focal length 0.090909
Ratio of displacement to focal length 0.090909
Ratio of displacement to focal length 0.090909
Ratio of displacement to focal length 0.090909

Published with MATLAB® R2024a