Exercise 5.4: The diffraction limit of a telescope Our ability to resolve detail in astronomical observations is limited by the diffraction of light in our telescopes. Light from stars can be treated effectively as coming from a point source at infinity. When such light, with wavelength A, passes through the circular aperture of a telescope (which we'll assume to have unit radius) and is focused by the consisting of central spot surrounded by a series of concentric rings. The intensity of telescope in the focal plane, it produces not a single dot, but a circular diffraction pattern the light in this diffraction pattern is given by where r is the distance in the focal plane from the center of the diffraction pattern, k = 277/A, and J₁ (x) is a Bessel function. The Bessel functions Jm (x) are given by Im (x) 2 1(r) = (h(kr)) ², = TT 7/7 kh cos(mex sin 0) de, where m is a nonnegative integer and x > 0. a) Write a Python function J (m, x) that calculates the value of Jm (x) using Simpson's rule with N = 1000 points. Use your function in a program to make a plot, on a single graph, of the Bessel functions Jo, J₁, and J2 as a function of x from x = 0 to x = 20. b) Make a second program that makes a density plot of the intensity of the circular diffraction pattern of a point light source with A = 500 nm, in a square region of the focal plane, using the formula given above. Your picture should cover values of r from zero up to about 1 μm.

Exercise 5.4: The diffraction limit of a telescope Our ability to resolve detail in astronomical observations is limited by the diffraction of light in our telescopes. Light from stars can be treated effectively as coming from a point source at infinity. When such light, with wavelength A, passes through the circular aperture of a telescope (which we'll assume to have unit radius) and is focused by the consisting of central spot surrounded by a series of concentric rings. The intensity of telescope in the focal plane, it produces not a single dot, but a circular diffraction pattern the light in this diffraction pattern is given by where r is the distance in the focal plane from the center of the diffraction pattern, k = 277/A, and J₁ (x) is a Bessel function. The Bessel functions Jm (x) are given by Im (x) 2 1(r) = (h(kr)) ², = TT 7/7 kh cos(mex sin 0) de, where m is a nonnegative integer and x > 0. a) Write a Python function J (m, x) that calculates the value of Jm (x) using Simpson's rule with N = 1000 points. Use your function in a program to make a plot, on a single graph, of the Bessel functions Jo, J₁, and J2 as a function of x from x = 0 to x = 20. b) Make a second program that makes a density plot of the intensity of the circular diffraction pattern of a point light source with A = 500 nm, in a square region of the focal plane, using the formula given above. Your picture should cover values of r from zero up to about 1 μm.