In the chapter 4 of the text "Introduction To Fourier Optics" by Goodman, the diffraction pattern of light passing through a circular aperture was studied. The diffraction pattern on the image plane $(x_i, y_i)$, is given by the fourier transform of the amplitude transmittance $U(x_o,y_o)$ on the aperture plane $(x_o, y_o)$ with relevant phases; through the Fraunhofer approximation. I.e.,

$$U(x_i,y_i) = \frac{\exp[ikz] \exp[ik(x_i^2 + y_i^2)/2z]}{i\lambda z}\mathcal{F}\left[U(x_o,y_o)\right]\, .$$

For a circular aperture of diameter $l$, the author uses the amplitude transmittance, $U(x_o,y_o)=circ\left(\frac{r}{l/2}\right)$. Where $r=\sqrt{x_o^2+y_o^2}$, and,

$$ circ(r) = \begin{cases} 1, & \text{for } r < 1 \\ 1/2, & \text{for } r = 1 \\ 0, & \text{for }r > 1 \, . \end{cases} $$

My question is, what is the reason for the $1/2$ when $r=1$? Are we not able to use a heaviside step function in the radial coordinate to encapsulate the amplitude transmittance? I.e. $U(x_o,y_o) = H(r-l/2)$, where,

$$ H(r-r_0) = \begin{cases} 1, & \text{for } r \leq r_0 \\ 0, & \text{for } r > r_0 \,. \end{cases} $$


A function that satisfies the so-called Dirichlet condition, that is one that has finite number of discontinuities and finite number of monotonic pieces (more generally has finite variation), see 1, will have its Fourier series or transform take the mid-point value at the points of discontinuities, so it is convenient to assume that value from the beginning since it makes no difference in the end.


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