4. A plane wave is incident on a circular aperture of radius \( a \). The Fresnel diffraction pattern due to the aperture is observed on a screen \( S \) placed at a distance \( z \). Calculation of the complete diffraction pattern using expression of the Fresnel diffraction is difficult and is avoided. Therefore, find out the intensity variation as function of \( z \)-axis (here \( x=0 \) and \( y=0 \) ) in the Fresnel diffraction regime. Plot the intensity as function of \( z \).
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Fresnel diffraction is a type of wave diffraction that occurs when a wave passes through an aperture and then propagates to a region where the phase of the wave is not constant. This is in contrast to Fraunhofer diffraction, where the phase of the wave is constant Show more…
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Consider the Fresnel diffraction pattern produced by a plane wave incident normally on a slit of width b. Assume $\lambda=5 \times 10^{-3} \mathrm{~cm}, d=100 \mathrm{~cm} .$ Using Table $20.1$, approximately calculate the intensity values (for $b=0.1 \mathrm{~cm}$ ) at $\mathrm{y}$ $=0, \pm 0.05 \mathrm{~cm}, \pm 0.1 \mathrm{~cm}$. Ropeat the analysis for $b=2 \mathrm{~cm} .$
Consider the Fresnel diffraction pattern produced by a plane wave incident normally on a slit of width $b$. Assume $\lambda=5 \times 10^{-5} \mathrm{~cm}, d=100 \mathrm{~cm} .$ Using Table $20.1$, approximately calculate the intensity values (for $b=0.1 \mathrm{~cm}$ ) at $y$ $=0, \pm 0.05 \mathrm{~cm}, \pm 0.1 \mathrm{~cm}$. Repeat the analysis for $b=5 \mathrm{~cm}$.
Light waves of frequency $\lambda$ passing through a slit of width $a$ produce a Fraunhofer diffraction pattern of light and dark fringes (Figure 16). The intensity as a function of the angle $\theta$ is \begin{equation}I(\theta)=I_{m}\left(\frac{\sin (R \sin \theta)}{R \sin \theta}\right)^{2}\end{equation} where $R=\pi a / \lambda$ and $I_{m}$ is a constant. Show that the intensity function is not defined at $\theta=0 .$ Then choose any two values for $R$ and check numerically that $I(\theta)$ approaches $I_{m}$ as $\theta \rightarrow 0$ .
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Limits: A Numerical and Graphical Approach
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