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Math Help - Light spreading through different mediums

  1. #1
    Senior Member TriKri's Avatar
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    Light spreading through different mediums

    Hello!

    I just have a simple question about light spreading where the light goes through different mediums with different refractive indexes. If the light travels from the light source (origo) with a speed of v, say in a two dimensional space, it will have reached the circle x^2 + y^2\ =\ v^2\cdot t^2 after the time t.

    But if we add a line to the space that separates the first medium from a second medium, with a different refractive indexe and hence a different speed of light, the light spreading won't be in the shape of a circle anymore, it will be more complex! Is there any equation as a function of time that can explain how the light will spread after entering the second medium?
    Last edited by TriKri; June 14th 2008 at 01:19 AM.
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by TriKri View Post
    Hello!

    I just have a simple question about light spreading where the light goes through different mediums with different refractive indexes. If the light travels from the light source (origo) with a speed of v, say in a two dimensional space, it will have reached the circle x^2 + y^2\ =\ v\cdot t after the time t.

    But if we add a line to the space that separates the first medium from a second medium, with a different refractive indexe and hence a different speed of light, the light spreading won't be in the shape of a circle anymore, it will be more complex! Is there any equation as a function of time that can explain how the light will spread after entering the second medium?
    Actually your circle equation will be:
    x^2 + y^2 =v^2 t^2

    When light enters a medium (ie. something that isn't free space) it "slows down." (For you relativity buffs, yes it still travels at c. But it takes time for it to be absorbed, then re-emitted from atom to atom so its net speed is less than c.) The overall effect of this is something called the "index of refraction" of the material, usually denoted as the unitless constant n > 1. The speed of light in a material of index of refraction n is:
    v = \frac{c}{n}

    The problem with what you are asking is that every point on the wavefront you initially described undergoes this speed change when at the point where the wavefront hits the new medium. New wavelets are generated at all points along the boundary. The description of the effect is called "Huygen's Principle."

    It is easier to see what happens with linear wavefronts, as opposed to circular ones. When the line wavefronts hit the new boundary a new line wavefront is generated (as a result of Huygen's Principle). A similar effect occurs for circular wavefronts, they produce new circular wavefronts. (In general the radius of the new circular fronts is different from the one in the source medium, and the center is also changed.)

    I haven't found a good internet source to help you with this, but the new propagation speed of light can be found if you know the index of refraction of the material, and you can trace the center of the new circular fronts by "winding time" backward in your equation. (I don't have the equation at my fingertips, and don't have my source for this with me. But it should be relatively easy for you to write the equations out.)

    -Dan
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    Senior Member TriKri's Avatar
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    So it's just a new circle? That's great! I'm serious; it really made the actual problem a lot easier now. Actually I thought it would be some sort of shape that looked like a circle, but I wasn't sure whether it was a new circle equation or if it was more complex ... anyway, thanks.

    And oh, the equation of a circle, just a little mistake writing vt instead of v^2t^2... hehe.

    I guess it's difficult to derive the fact that it will just remain the equation of a circle. Anyway, I have to try to derive it! Maybe I’ll put the derivation here if I succeed to make it.
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by TriKri View Post
    So it's just a new circle? That's great! I'm serious; it really made the actual problem a lot easier now. Actually I thought it would be some sort of shape that looked like a circle, but I wasn't sure whether it was a new circle equation or if it was more complex ... anyway, thanks.

    And oh, the equation of a circle, just a little mistake writing vt instead of v^2t^2... hehe.

    I guess it's difficult to derive the fact that it will just remain the equation of a circle. Anyway, I have to try to derive it! Maybe I’ll put the derivation here if I succeed to make it.
    Feel free, and good luck! It's quite possible to do, but I don't recall how its done.

    -Dan

    PS I'd start with the linear wavefronts generating another linear wavefront in the new medium. That one is easy to prove.
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  5. #5
    Senior Member TriKri's Avatar
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    Hm, I made numerical calculations of the photons positions in Exel, It seems like they're not in a circular shape...


    Thanks anyway.
    Last edited by ThePerfectHacker; June 20th 2007 at 12:09 PM.
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by TriKri View Post
    Hm, I made numerical calculations of the photons positions in Exel, It seems like they're not in a circular shape...


    Thanks anyway.
    As I said, it's not easy. Try doing it for linear wavefronts. It should be easy to visualize why the new wavefronts are linear in the new material. Once you see what's going on with that then try your algorithm in Excel and see if it will produce the linear waves. If not, then you've got a glitch.

    -Dan
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  7. #7
    Senior Member TriKri's Avatar
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    This animated image on wikipedia illustrates that circular wave fronts entering a new media do not spread like circles anymore:



    -Kristofer
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