Fresnel diffraction from square: on axis intensity

In summary, the amplitude of the pattern with the square is reduced by a factor and the intensity is increased by a factor.
  • #1
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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  • #2
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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  • #3
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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FAQ: Fresnel diffraction from square: on axis intensity

What is Fresnel diffraction from square?

Fresnel diffraction is a phenomenon that occurs when light waves pass through an aperture or around an object, causing interference and producing a pattern of light and dark regions. In the case of a square aperture, the diffraction pattern will be similar to that of a single slit, but with additional features due to the square shape.

How is the on-axis intensity of a square aperture affected by Fresnel diffraction?

The on-axis intensity of a square aperture will exhibit a central bright spot, similar to that of a single slit diffraction pattern, but with additional bright and dark regions due to the square shape. The intensity will also decrease as the distance from the center increases, creating a series of concentric rings.

What factors affect the on-axis intensity in Fresnel diffraction from a square aperture?

The on-axis intensity in Fresnel diffraction from a square aperture is affected by the size of the aperture, the wavelength of the light, and the distance between the aperture and the observation point. The shape and orientation of the aperture can also have an impact on the intensity distribution.

How does the size of the square aperture affect the diffraction pattern?

The size of the square aperture has a direct impact on the diffraction pattern. As the aperture size increases, the central bright spot becomes larger and the intensity decreases at a slower rate as the distance from the center increases. A smaller aperture will produce a more pronounced diffraction pattern with a smaller central bright spot and faster decrease in intensity.

What is the difference between Fresnel diffraction and Fraunhofer diffraction?

Fresnel diffraction occurs when the light waves pass through an aperture or around an object at a close distance, while Fraunhofer diffraction occurs at a much larger distance. In Fresnel diffraction, the light waves are still curved and interfere with each other, creating a complex diffraction pattern. In Fraunhofer diffraction, the light waves are almost parallel and the diffraction pattern is simpler, resembling a single slit pattern.

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