I have read some papers about wave-front measurement, and measuring the spherical aberration. But didn't say any thing about the method!
How the measuring of wave-front, relate to the measure of "zernike aberration coefficients value"?! What is the the dimension of these coefficients? I saw that the dimension is length, and the results reported in "um"!
In a ideal imaging system, all rays reaching the image plane from a point in the object plane have identical optical path lengths at the conjugate point in the image plane. Wavefront distortion is the difference in optical path length from some reference, measured along the direction of propagation. The measurement surface is typically a plane oriented normal to the principal axis (or the direction of propagation), and close to the imaging lens or mirror.
The Zernike coefficients are found by fitting Zernike polynomials to the measured distortion over the measurement surface. Note that there are various schemes for labelling and normalising Zernike polynomials, and this will change the magnitude of the reported coefficients. ANSI Z80.28-2004 is now widely used by the ophthalmic community, but other conventions are used elsewhere.
Methods for measuring aberrations include interferometric techniques, which measure phase difference directly, Shack-Hartmann sensors, which measure wavefront slope, and curvature sensors which measure the rate of change of intensity in the the direction of propagation, at points across the measurement surface. There are also phase recovery schemes which compare the intensity distributions in two or more planes at or near the image plane and with different amounts of defocus.
A uniform phase shift has no effect on intensity or image quality, so the zero order Z(0,0) term is conventionally set to zero or ignored. First order Zernike terms describe wavefront tilt, and cause a transverse shift in image position. Second order Z(2,0), Z(2,+2), Z(2,-2) are quadratic distortions describing defocus and astigmatism at 0 and 45 degrees. Other terms describe higher order aberrations.
http://voi.opt.uh.edu/VOI/WavefrontCongress/2005/presentations/2-Campbell_ANSIstd.pdf
http://voi.opt.uh.edu/3_Schwiegerling%20ANSI.pdf
http://webstore.ansi.org/RecordDetail.aspx?sku=ANSI+Z80.28-2010
I have few to add to what Alan just answered. But I think you need a practical answer.
In practice you have measured a wavefront let's say W(x,y) where W is the deviation from perfect planarity (expressed usually in µm). First you should limit your analysis on a circular aperture. So If you have a square Wavefront just cut a circular piece of it. If you want to know for example how much Coma along y your Wavefront has, that is the Zernike coefficient for the component 2 ,+2, all you have to do is to take the expression of Z(2,+2) (x,y), normalize it on the Wavefront circular aperture (that is integrate Z(2,+2)(x,y) two times on the aperture and make the square root you get a coefficent C and your Coma is then 1/C*Z(2,+2)=Z'(2,+2)) and finally make the integral of the product W(x,y)*Z'(2,+2) on all the circular aperture. The result is the coefficient you are looking for. To exctract the oter coefficients just do the same procedure.The fact is that Zernike Polynomials are orthonormal and every Wavefront canbe seen as superposition of these polynomials with different coefficients. In addition I can suggest you this presentation online: http://www.dm.unibo.it/home/citti/html/AnalisiMM/Schwiegerlink-Slides-Zernike.pdf......you have to carve your way through.
Sorry I want to make precise a point in my answer:....to find the normalization coefficient you have to integrate on the aperture the product Z(2,+2)(x,y)*Z(2,+2)(x,y) that is the square of the polynomial and then make the square root of the result.
Gabriele: Coma is described by Z(3,+/-1), not Z(2,+2) - in other words. Z31 and Z3-1, (or by Z7 and Z8 in the ANSI single index scheme).
Spherical aberration is Z40 (Z12 in the ANSI single index scheme).
The ANSI standard recommends that where possible the dual index (Zradial, azimuthal order notation) should be used in preference to the single index scheme, unless a 1-dimensional vector of coefficients is required.
An explicit formula for the normalization factor Nnm is given on slide 14 of the presentation you referenced. A corrected link is below.
http://www.dm.unibo.it/home/citti/html/AnalisiMM/Schwiegerlink-Slides-Zernike.pdf
Right thanks Alan I confused with Astigmatism but the procedure was anyway clear I hope. And yes there's an explicit formula for the normalisation you are right, it will make the computation faster instead of numerically extrapolate it.
The Zernike's polynoms coefficients are not measured per se. They are obtained by fitting a set of Zernike's polynomials to the data.
Be aware that the usual Zernike's polynomials are valid only for unobscured circular apertures. For circular apertures with obscuration you have to use a set of tailored Zernike's polynomials which you can find in litterature. For rectangular apertures you may want to use Legendre's polynomials. More generally you want to chose polynomials that meet the orthogonality condition over the geometry of the aperture of interest.
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