Fresnel diffraction from square: on axis intensity

AI Thread Summary
The discussion focuses on verifying the correctness of a mathematical derivation related to Fresnel diffraction from a square aperture. The author outlines the separation of the Fresnel integral using symmetry and substitutions to simplify the calculations. They derive an expression for the amplitude of the diffraction pattern and its intensity, noting a reduction factor due to the square aperture. A key point of contention arises regarding the values of C(∞) and S(∞), which the author initially misstates as equal to 1, but later acknowledges they are actually 1/2. The conversation concludes with a correction regarding the Cornu spiral, confirming the adjustments made to the calculations.
ergospherical
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Homework Statement
Determine the factor by which the on-axis intensity from a square aperture of side length ##a## is reduced, compared to an unobstructed pattern.
Relevant Equations
Fresnel integral.
I'd appreciate if someone could check whether my work is correct. The ##x##-##y## symmetry of the aperture separates the Fresnel integral:\begin{align*}
a_p \propto \int_{-a/2}^{a/2} \mathrm{exp}\left(\frac{ikx^2}{2R} \right) dx \int_{-a/2}^{a/2} \mathrm{exp}\left(\frac{iky^2}{2R} \right) dy \equiv \left[ \int_{-a/2}^{a/2} \mathrm{exp}\left(\frac{ikx^2}{2R} \right) dx \right]^2
\end{align*}I substitute ##\frac{\pi \xi^2}{2} = \frac{kx^2}{2R}##, i.e. ##\xi = x\sqrt{\frac{k}{\pi R}}##. I also denote ##w = \frac{a}{2} \sqrt{\frac{k}{\pi R}}## to simplify the new limit,\begin{align*}
a_p \propto \frac{\pi R}{k} \left[ \int_{-w}^{w} \mathrm{exp}\left(\frac{i\pi \xi^2}{2} \right) d\xi \right]^2 = \frac{4\pi R}{k} \left[ \int_{0}^{w} \mathrm{exp}\left(\frac{i\pi \xi^2}{2} \right) d\xi \right]^2
\end{align*}and separate into sines and cosines,\begin{align*}
a_p \propto \frac{4\pi R}{k} \left[ \int_{0}^{w} \cos{\left(\frac{\pi \xi^2}{2} \right)} d\xi + i\int_{0}^{w} \sin{\left(\frac{\pi \xi^2}{2} \right)} d\xi \right]^2 = \frac{4\pi R}{k} \left[ C\left( w \right) + iS\left( w \right) \right]^2
\end{align*}The unobstructed pattern would instead correspond to ##w \rightarrow \infty##, in which case ##[C(\infty) + iS(\infty)]^2 = [1 + i]^2 = 2i##. The amplitude of the pattern with the square is therefore reduced by a factor\begin{align*}
\alpha = \frac{\left[ C\left( w \right) + iS\left( w \right) \right]^2}{2i}
\end{align*}and the intensity by ##|\alpha|^2##,\begin{align*}
|\alpha|^2 = \frac{|C\left( w \right) + iS\left( w \right) |^4}{4}
\end{align*}Does it look correct?
 
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The only mistake that I see is where you wrote

ergospherical said:
##[C(\infty) + iS(\infty)]^2 = [1 + i]^2 = 2i##.

##C(\infty)## and ##S(\infty) ## are not equal to 1. They equal 1/2.
 
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Likes PhDeezNutz and ergospherical
TSny said:
##C(\infty)## and ##S(\infty) ## are not equal to 1. They equal 1/2.
Ah great, thanks, I’d mis-remembered the Cornu spiral.
 
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