A quick implementation of of Zernike polynomials for modeling wavefront aberrations to to optical turbulence. Equations pulled from Michael Roggemann's Imaging Through Turbulence.
Zernike polynomials are often used to model wavefront aberrations for various optics problems. The equations are expressed in polar coordinates, so to calculate the image we first convert a grid into polar coordinates using the relation
And then calculate the Zernike Polynomials as
Where the functions are called Radial Functions are are calculated as
With Noll normalization. Certain polynomials have names. First is , know as Piston as is often subtracted to study of single aperture systems. Next, and are ramps that are called tilt and cause displacement. is a centered ring structure known as defocus while and are astigmatism, and are coma, and is spherical aberration.
They can be imaged as follows:
All created using the following code:
function result = z_cart(i,m,n, x,y) %Z_POL result the zernike polynomial specified in cartesian coordinates % calculate the polar coords of each point and use z_pol: r = (x.^2 + y.^2).^(1/2); thta = atan2(y,x); result = z_pol(i,m,n,r,thta); end function result = z_pol(i, m,n, r,t) %Z_POL result the zernike polynomial specified in polar coordinates if(m==0) result = radial_fun(0,n,r); else if mod(i,2) == 0 % is even term2 = cos(m*t); else term2 = sin(m*t); end result = sqrt(n+1)*radial_fun(m,n,r).*term2; end end function result = radial_fun(m,n,r) %RADIAL_FUN calculate the radial function values for m,n, and r specified result = 0; for s = 0:(n-m)/2 result = result + (-1)^s *factorial(n-s)/ ... (factorial(s)*factorial((n+m)/2 -s)*factorial((n-m)/2-s)) *... r.^(n-2*s); end end