- Monday, April 16, at noon
- The finished Scilab code and analysis

Instructor's solution: rect_integ.sci |

Instructor's solution: trap_integ.sci |

Instructor's solution: simp_integ.sci |

Your job in this project: write programs to

- integrate a given function of one variable over a given interval, using three different methods
- analyze properties of the diffraction pattern when light passes through a rectangular slit, and when light passes through a circular aperture

You may find some notes of the properties of light and diffraction in lecture notes on diffraction.

You may **not** use Scilab's built-in functions for
finding maxima or performing numerical integration -- instead,
please implement these algorithms yourself.

You'll need to modify some of your existing Scilab source code files from last week in order to write two new functions. Don't worry -- they are all very much alike, and you'll be able to cut and paste.

- rect_integ.sci
- trap_integ.sci
- simp_integ.sci

and then we compute

Your job is to integrate this function over various ranges of
angle **θ**.
You may use the sinc_squared.sci
function I wrote, or one of your own.

You must provide three (very similar) functions this week:

function y = rect_integ(start_x, end_x, epsilon, func) function y = trap_integ(start_x, end_x, epsilon, func) function y = simp_integ(start_x, end_x, epsilon, func)where the arguments are

start_x start of the interval to integrate end_x end of the interval to integrate epsilon termination criterion: stop iterating when the fractional change in integral drops to this level func the function to integrate y (output argument) is the result of the integration

Each routine integrates the given function over the given interval and returns as its output argument the result. For example, if I provided the simple function simple_func.sci

function y = simple_func(x) y = 2*x; endfunction

and then called the *rect_integ* function like so:

a = rect_integ(1, 3, 1e-4, simple_func);

I should end up with **a = 7.9995... **.

I suggest that you test your functions by using them to integrate simple functions, to which you know the correct answer.

Questions about the numerical algorithms:

- Set the wavelength
**lambda = 1.0**and the width of the slit**a = 10.0**. Use each method (rectangular, trapezoidal, Simpson's) to integrate the intensity of the diffraction pattern from an angle**θ = 0**to**θ = 0.3**. For each of the three methods, you must divide the interval into 16, 32, 64, 128, 256 little pieces. Make a table which shows for each method# of slices fractional change since previous iteration ---------------------------------------------------------- 16 32 64 128 256

- Using your tables as evidence, answer the following question:
- Which method converges most rapidly?
- How fast does each method converge? That is, if you double the number of slices, does the fractional change go down by a factor of 2? Or a factor of 4? Or what?

- For each method, find out how many pieces into which you must break the interval in order to reach a fractional change of 1e-5. Make a table which shows the number the pieces, and the time it took your program to reach this point and evaluate the integral, for each method. That is, include the time it takes your program to iterate until it reaches the number of pieces required to satisfy the criterion. Which is fastest?

Let's now ask a few questions about the physics --
properties of the pattern of light itself.
You may use whichever method you wish to answer them.
Continue to use **lambda = 1** and **a = 10**.

- At what angle does the first dark spot occur? That is, at what angle will the light intensity first drop to zero?
- At what angle does the second dark spot occur?
- At what angle does the third dark spot occur?
- What fraction of the total light falls in the central bright spot?
- What fraction of the total light falls in the second bright spot?
- What fraction of the total light falls in the third bright spot?

*Bells and Whistles*

- Using the same conditions as in question 1 above, make a graph which shows the fractional change on the x-axis, and the number of iterations required to find the integral to that precision on the y-axis. Use logarithmic axes in both directions. Plot the points for each method together on the same graph, so that you can compare the methods directly. It's okay if some methods have a different range than other methods (that is, the rectangle method may only go to epsilon = 1e-3, but the trapezoid method to epsilon = 1e-5, or something like that). What does the graph tell you about the efficiency of the different methods?
- Light which passes through a circular aperture
will show a bright spot at the center
of the diffraction pattern, surrounded
by rings which gradually fade away.
The mathematical description of the intensity of light
as a function of angle away from the center is called
the Airy function, after the Astronomer Royal who
first analyzed it.
I've written a routine,
airy_func.sci,
which will give you the intensity as a function of angle:
function y = airy_func(lambda, diam, theta)

Use this function, and one of your routines, to simulate starlight focused by the Hubble Space Telescope, which has a diameter of**2.4 meters**. Adopt a wavelength in the middle of the visible range,**lambda = 550 nm**.- Make a graph showing the intensity as a function
of angle, with limits
**theta = 0**to**theta = 10 microradians**. - The resolution of HST is set roughly by the point in this diffraction pattern at which the intensity first falls to zero. What angle is this?
- Suppose HST takes a picture of one of the Apollo landing sites on the Moon. Using your answer to the previous question, estimate the linear size (in meters) of the smallest object that HST could resolve. Could HST detect the base of the lunar module which still sits on the lunar surface?

- Make a graph showing the intensity as a function
of angle, with limits

*
This page maintained by Michael Richmond.
Last modified Apr 10, 2007.
*

Copyright © Michael Richmond. This work is licensed under a Creative Commons License.