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Wave optics is cool. You don't need to know the law of reflection - the waves know from their self where they have to go.
They only thing you need to know is
- every point of a wave front emittes a wavelet (that is a spherical wave)
- wavelet interfere (at every point), constructive or destructive
- the speed of light in glass or another medium is c/n. n is the refractive index
We don't need the last rule for this calculation here.
The main idea behind this is that every point generates a wavelet, which is a spherical wave. This waves goes to everywhere. There are places where more waves met and increase each other, and other places where they are lost by destructive interference. All you have to do is to measure the optical path of each wavelet.
I started with the program of a slit. Now I came to a more important example: The calculation of the Airy disc. I started (again with something simple), with a parabloid mirror like used in a Newtonian telescope.
If you try to understand how the program works I recommend consulting the accompanying sketch.
In front of the mirror we have / we assume a flat and coherent wave front. All rays here are parallel. The optical path need is the amount of "g" to the mirror and then the amount of "q" to the screen. From everywhere of the mirror's surface goes a wavalet which hit the srceen at the actual (red dot) place. This rays interfere.
My calculation assumes that the central (blue) ray has always a maximum amplitude. The phase shift of all the other rays calculates form the differences in the optical path.
The optical path of the blue ray is the line form the incoming wave to the mirror G and the line Q form the center of the parabloid to our point. The value of G is the so called the mirrors sagitta, which can be calculated by
G = A²/(4*f), where A is the aperture and f the focal length.
For the optical path of the green ray we need the value of g. This is the difference between the sagitta and thy y coordinate of the mirror. The y coordinate depends form the distance to the center of the mirror, which is r in my program. I first calculate the "height" of the mirror, which is y = x²/4f. In principle, this is the same formula which I used for the sagitta.
The difference between the tracking of the rays is f. Multiplied with 2*pi whe get the phase angle of the wavelet at the screen.
The summation of the amplitude values does not give the intensity of the light at the places of the screen. The amplitude value needs to be squared for the intensity. The reason is, that light is an electromagnetic wave which consists of an electric and a magnetic component. The intensity is the Cartesian product of them, the so called Poynting vector.
I gave the program in the simplest form. You may experiment with it and try, whether the diameter of the airy disc and the intensity of the first ring follows the exceptions. The Airy disc has the size 2.44 * lambda * f / A, the first ring has an intensity of 1.75% independent of the Aperture and focal length. The values are not very accurate in the given program. But if you increase the number of points in each direction to 30 or even 100, all things look much better. The calculator runs for a few hours then.
You may experimenting further: What happens if the mirror has a central obstruction (which it has in most cases)? Simply modify the line If r<=A/2 . What happens if you are not in focus? Just add something to the Y term in the Q,q formula and increase the width of the screen w. You get a diffraction pattern then, sich patterns are used for testing optics.
You may experiment further and look what happens with an oblique wave front - just add and subtract something to the values of g. The problem is not symmetric anymore and you need to calculate a real image plane. not just a line. This takes a while with the pocket calculator.