From ed96bc6c0dff2ebcefa9f64f2fdcd496b6428aea Mon Sep 17 00:00:00 2001 From: Ceres Date: Mon, 1 Dec 2025 18:01:35 +0000 Subject: [PATCH] Update Exercise 2 --- Exercise 1/main.pdf | Bin 906612 -> 906612 bytes Exercise 1/main.tex | 1 + Exercise 2/.chktexrc | 4 ++ Exercise 2/exercise2.py | 147 ++++++++++++++++++++++++++++------------ Exercise 2/main.pdf | Bin 0 -> 137577 bytes Exercise 2/main.tex | 91 +++++++++++++++++++++++++ 6 files changed, 200 insertions(+), 43 deletions(-) create mode 100644 Exercise 2/.chktexrc create mode 100644 Exercise 2/main.pdf create mode 100644 Exercise 2/main.tex diff --git a/Exercise 1/main.pdf b/Exercise 1/main.pdf index 20bce0634505172432bf3ac7d252a528a9be8651..a47640ccc207f353cb0d9abc2bd85ff7e9b75723 100644 GIT binary patch delta 154 zcmeye%KXbJ^M)427N!>F7M2#)7Pc1l7LFFq7OocV7M?A<2hOS)85kOxo0=I|L}_yA z`{t*(B$lKqXt-Dz85o%w!X>vqJIi~K#o55z+0n(t#Ld#e)Wy)$+|b3)+``G((!#{p W$il_d#MDl~hLDo&zb^4sF#!PiR4a@C delta 154 zcmeye%KXbJ^M)427N!>F7M2#)7Pc1l7LFFq7OocV7M?A<2hOS)8k!ke7@C=wMrm^C z`{t*(B$lKqXt-Dz85o%w!X>vqJIi~K#o5i(($vk&)y>(&)YQ_%(Z$fs!qM2t#LU#f W(Z$lm$ aperture: + if (xp[i]**2+yp[i]**2) > (aperture/2)**2: values.append(0) else: - value = np.exp(((1j*k)/(2*z))*((x-xp)**2+(y-yp)**2)) - values.append(value.real) + value = np.exp(((1j*k)/(2*z))*((x-xp[i])**2+(y-yp[i])**2)) + values.append(value.imag) return np.array(values) def monteCarlo(x, y, z, k, aperture): @@ -126,30 +138,79 @@ def monte(): error = aperture*np.sqrt((meansq-mean*mean)/N) return integral, error - def genData(aperture, z, k, screen_range): + def genData(aperture, z, k, resolution, screen_range): - xs = np.linspace(-screen_range/2, screen_range/2, num=50) - ys = np.linspace(-screen_range/2, screen_range/2, num=50) + xs = np.linspace(-screen_range/2, screen_range/2, num=resolution) + ys = np.linspace(-screen_range/2, screen_range/2, num=resolution) - constant = k/2*np.pi*z intensities = [] + completion = 0 - for y in ys: + constant = k/(2*np.pi*z) + + for y in tqdm(ys): xIntensities = [] - for x in xs: + for x in tqdm(xs): integral, error = monteCarlo(x, y, z, k, aperture) I = c*e0*constant*integral + if I < 0.005: + I = 0 + # completion = completion + 100/resolution**2 + # print(completion) xIntensities.append(I) intensities.append(xIntensities) intensities = np.array(intensities) return intensities - intensity = genData(2e-4, 0.005, 8.377e6, 0.002) - extents = (-0.001,0.001,-0.001,0.001) + intensity = genData(aperture, z, k, resolution, screen_range) + extents = (-screen_range/2,screen_range/2,-screen_range/2,screen_range/2) - plt.imshow(intensity,vmin=0.0,vmax=1.0*intensity.max(),extent=extents,origin="lower",cmap="nipy_spectral_r") + plt.imshow(intensity,vmin=1.0*intensity.min(),vmax=1.0*intensity.max(),extent=extents,origin="lower",cmap="nipy_spectral_r") plt.colorbar() plt.show() -monte() +MyInput = '0' +while MyInput != 'q': + MyInput = input('Enter a choice, "1", "2", "3", "4" or "q" to quit: ') + print('You entered the choice: ',MyInput) + if MyInput == '1': + print('You have chosen part (1): 1D rectangular diffraction') + aperture = input("Please input the desired aperture (m): ") + z = input("Please enter the desired distnce from the screen (m): ") + wl = input("Please enter the desired wavelength of light (m): ") + k = (2*np.pi)/float(wl) + screen_range = input("Please enter the diameter of the screen (m): ") + resolution = input("Please enter the resolution of the plot (pixels): ") + plot1D(float(aperture), float(z), float(k), float(screen_range), int(resolution)) + elif MyInput == '2': + print('You have chosen part (2): 2D rectangular diffraction') + aperture = input("Please input the desired aperture (m): ") + z = input("Please enter the desired distnce from the screen (m): ") + wl = input("Please enter the desired wavelength of light (m): ") + k = (2*np.pi)/float(wl) + screen_range = input("Please enter the diameter of the screen (m): ") + resolution = input("Please enter the resolution of the plot (pixels): ") + plot2Drectangular(float(aperture), float(z), float(k), float(screen_range), int(resolution)) + elif MyInput == '3': + print('You have chosen part (3): 2D circular diffraction') + aperture = input("Please input the desired aperture (m): ") + z = input("Please enter the desired distnce from the screen (m): ") + wl = input("Please enter the desired wavelength of light (m): ") + k = (2*np.pi)/float(wl) + screen_range = input("Please enter the diameter of the screen (m): ") + resolution = input("Please enter the resolution of the plot (pixels): ") + plot2Dcircular(float(aperture), float(z), float(k), float(screen_range), int(resolution)) + elif MyInput == '4': + print('You have chosen part (3): 2D circular diffraction usinf Monte Carlo') + aperture = input("Please input the desired aperture (m): ") + z = input("Please enter the desired distnce from the screen (m): ") + wl = input("Please enter the desired wavelength of light (m): ") + k = (2*np.pi)/float(wl) + screen_range = input("Please enter the diameter of the screen (m): ") + resolution = input("Please enter the resolution of the plot (pixels): ") + samples = input("Please enter the desired nu,ber of samples for the calculation: ") + monte(float(aperture), float(z), float(k), float(screen_range), int(resolution), int(samples)) + elif MyInput != 'q': + print('This is not a valid choice') +print('You have chosen to finish - goodbye.') diff --git a/Exercise 2/main.pdf b/Exercise 2/main.pdf new file mode 100644 index 0000000000000000000000000000000000000000..74f2491fcb6943b5a0be7236ece575877c33fd17 GIT binary patch literal 137577 zcma&tLy#~$*e>X{ZQHhO+qP}nwr$(C&9`ma?mpj4%|A6&XBH=`C!6e&+^H)Mse*_Y zEh8N(6zRgs=szeXRsseBdm}3-9v&!q8B;rR7YhO=77hl2|9e2ui&@&Zm^u;Ai`f{u zn2MMh+nbm|@$o@9yEvH|+Cq73-e~DMZ?~cPuGjBtcP`Tf*(Y2)7dl?ca2;i?sg^U# zGAUc7uQ|!wP?4;+Jzja!0XR*vwrWjEC4=`z5I8tGI^KfARh-B`nA;A)&&O2rSk{f} zCKs~r>3;hRBIjjnGhEmlI?XXbu$Vr+9^zvRZ69|-y9ntR>4kywIvX=%tIWd-zA$_> 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Two calculation methods were used, a numerical calculation and Monte Carlo integration. The numerical method gave cleaner, less `noisy' data, but the Monte Carlo method was considerably quicker. + +\end{abstract} + +\section{Introduction} + +In this exersise we will use numerical methods to investigate the diffraction of light when passed through an aperture of different shapes. We will use two methods, first pure numerical integration methods, and then the Monte Carlo method, and compare the accuraccy and speed of these two methods. We will first create a 1 dimensional plot of the diffraction to ensure the results are as expected, and then 2 dimensional plots, using both square and curcular apertures. + +\section{Theory and Methods} + +To calculate the diffraction pattern of light passing through a single aperture, we will use the Fresnel diffraction integral, given by: +\begin{equation} + E(x, y, z)=\frac{i^{ikz}}{i\lambda z}\int_{-\infty}^\infty\int_{-\infty}^\infty E(x^\prime,y^\prime)\text{exp}\bigg\{\frac{ik}{2z}\big[(x-x^\prime)^2+(y-y^\prime)\big]\bigg\}\mathrm{d}x^\prime\mathrm{d}y^\prime +\end{equation} +To allow for this to be calculated computtionally, we will simplify it, integrating over the area of the aperture, and seperating out the real and imaginary parts of the equation: +\begin{align} + E(x,y,z)=\frac{kE_0}{2\pi z}\int_{x_1^\prime}^{x_2^\prime}\int_{y_1^\prime(x^\prime)}^{y_2^\prime(x^\prime)}\cos\bigg\{\frac{k}{2z}\Big[(x-x^\prime)^2+(y-y^\prime)^2\Big]\bigg\}\mathrm{d}x^\prime\mathrm{d}y^\prime \nonumber \\ + +i\frac{kE_0}{2\pi z}\int_{x_1^\prime}^{x_2^\prime}\int_{y_1^\prime(x^\prime)}^{y_2^\prime(x^\prime)}\sin\bigg\{\frac{k}{2z}\Big[(x-x^\prime)^2+(y-y^\prime)^2\Big]\bigg\}\mathrm{d}x^\prime\mathrm{d}y^\prime +\end{align} + +\section{Explanation of Code} + +\section{Results and Discussion} + +\section{Conclusion} +\end{document}