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.
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Transcribed Image Text: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
2
1(r) = (h(kr)) ²,
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
cos(mex sin 0) de,
1
(x) kh
==
T
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.
-
Hint 1: You may find it useful to know that limx→0 J1(x)/x =
spot in the diffraction pattern is so bright that it may be difficult to see the rings around
1. Hint 2: The central
it on the computer screen. If you run into this problem a simple way to deal with it is to
use one of the other color schemes for density plots described in Section 3.3. The "hot"
scheme works well. For a more sophisticated solution to the problem, the imshow func-
tion has an additional argument vmax that allows you to set the value that corresponds
to the brightest point in the plot. For instance, if you say "imshow (x, vmax=0.1)", then
elements in x with value 0.1, or any greater value, will produce the brightest (most pos-
itive) color on the screen. By lowering the vmax value, you can reduce the total range of
values between the minimum and maximum brightness, and hence increase the sensi-
tivity of the plot, making subtle details visible. (There is also a vmin argument that can
be used to set the value that corresponds to the dimmest (most negative) color.) For this
exercise a value of vmax=0.01 appears to work well.
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