I was recently interested about a rigorous demonstration of the exact equivalence between angular spectrum propagation and Rayleigh-Sommerfeld propagation integrals, as mentioned in Goodman's "Introduction to Fourier Optics". I followed Goodman's reference for that: Sherman's "Application of the Convolution Theorem to Rayleigh's Integral Formulas" (1967)
In that publication, the second RS solution has a negative sign (equation 3), whereas in Goodman there is a positive sign. The positive sign seems logical from Goodman's demonstration with alternative Green's functions, but the negative sign leads to a perfect equivalence of the two RS solutions in the paper... Is there a mistake, or a missing step somewhere?
Hello Yuriy,
Yes, I understand that the choice of G- leads to the first RS solution with a negative sign, and the choice of G+ leads to the second RS solution with a positive sign in Goodman's “Introduction to Fourier Optics.”
However, in the paper from Sherman "Application of the Convolution Theorem to Rayleigh's Integral Formulas", both solutions are written with a negative sign. It's a bit disturbing since this paper aims at proving mathematically the equivalence between both solutions and the angular spectrum propagation method, and Goodman himself refers to that paper in these terms: "proved most elegantly by Sherman".
I can see one main difference between Goodman and Sherman: in Goodman, the derivative is with respect to n, in Sherman it is with respect to z (n being in the direction of the z axis for a planar surface normal to the direction of propagation along z, so it should not make a difference whether n or z).
Apparently Sherman took his equations from another paper (second ref., Bouwkamp 1954). Now looking at that paper, the two RS solutions do have a different sign... But they are reversed with respect to Goodman's! Goodman probably wanted to lay down the derivation of these equations in a clearer, tidier manner in his book in order to clarify the confusion about these signs.
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