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Zernike Polynomial

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

\(y = r\times sin(\theta) \)
\(r = \sqrt{x^2 + y^2} \)
\(\theta = tan^{-1}\frac{y}{x}\)

And then calculate the Zernike Polynomials as

\(Z_{i,even}(r,\theta) = \sqrt{n+1}R^m_n(r)cos(m\theta) \)
\(Z_{i,odd}(r,\theta) = \sqrt{n+1}R^m_n(r)sin(m\theta) \)
and \(Z_{i}(r,\theta) =R^0_n(r)\) if \(m=0\)

Where the functions \(R^m_n(r)\) are called Radial Functions are are calculated as

\(R^m_n(r) =\sum_{s=0}^{(n-m)/2} \frac{(-1)^s(n-s)!}{s![(n+m)/2 – s]![(n-m)/2 – s]!} r^{n-2s} \)

With Noll normalization. Certain polynomials have names. First is \(Z_1\), know as Piston as is often subtracted to study of single aperture systems. Next, \(Z_2\) and \(Z_3\) are ramps that are called tilt and cause displacement. \(Z_4\) is a centered ring structure known as defocus while \(Z_5\) and \(Z_6\) are astigmatism, \(Z_7\) and \(Z_8\) are coma, and \(Z_{11}\) is spherical aberration.

They can be imaged as follows:

Zernike Polynomial Z_13_4_2 Zernike Polynomial Z_1_0_0 Zernike Polynomial Z_2_1_1 Zernike Polynomial Z_3_1_1 Zernike Polynomial Z_4_2_0 Zernike Polynomial Z_5_2_2 Zernike Polynomial Z_6_2_2 Zernike Polynomial Z_7_3_1 Zernike Polynomial Z_8_3_1 Zernike Polynomial Z_9_3_3 Zernike Polynomial Z_10_3_3 Zernike Polynomial Z_11_4_0 Zernike Polynomial Z_12_4_2

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