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

and if

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

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