Separate the variable r01 for simplification. Factor the z^2 term in the formula for r01 so that it comes outside of the square root function. In short, the original formula, r01=sqrt(z^2+(x-ep)^2+(y-et)^2) becomes r01=z*sqrt(1+[(x-ep)^2/z]+[(y-et)^2/z]).
Apply binomial expansion to the square root term. The binomial expansion equation gives sqrt(1+[(x-ep)^2/z]+[(y-et)^2/z]) as an infinite sequence, 1 + .5*(x-ep)^2/z+.5*(y-et)^2/z+….
Replace the binomial expansion of the square root term into the equation for r01. This simplifies to r01=z[1+.5*(x-ep)^2/z+.5*(y-et)^2/z+…].
Delete all terms in the sequence. The equation for r01 further simplifies to r01=z[1+.5*(x-ep)^2/z+.5*(y-et)^2/z].
Plug r01 into the Fresnel diffraction integral and simplify. The result is U(x,y) = exp(jkz)/[j*theta*z]*Dint[U(ep,et)*exp(jk{[x-ep]^2+[y-et]^2}/(2z))], where “Dint” refers to the double integral with respect to et and ep.
Set up a function h(x,y) that will be used as a part of a convolution function for U(x,y). Let h(x,y) equal exp(jkz)/(j*theta*z)*exp{jk/(2z)*[(x-ep)^2+(y-et)^2)]}.
Replace all exp(jkz)/(j*theta*z)*exp{jk/(2z)*[(x-ep)^2+(y-et)^2)]} values in the Fresnal diffraction integral with h(x-ep,y-et).
Rewrite the integral in whole. The final, computable version of the Fresnel diffraction integral is Dint[U(ep,et)*h(x-ep,y-et)].
Use numerical software to compute the integral. Evaluate the integral with the boundary conditions from negative infinity to infinity. For example: In Maple, set the integral’s internal terms as a function with “f:= f(ep,et)*h(x-ep,y-et).” Make sure to input the numerical values for the variables as well as define the function h(x,y) in a similar manner. Then apply integration, with the int command, such as “int(f, x = -infinity..infinity).”