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Homework Help: A problem on the paraxial wave equation?

  1. Nov 28, 2009 #1
    I have this electrodynamics problem sheet on the paraxial approximation, and I am not getting very far with it. It starts off talking about a laser beam travelling in the z-direction, and says that a scalar wave has the form F(r,w)eiwt.

    The first part of the question ends with me proving that the homogeneous wave equation

    [tex]\nabla^2 \Psi = \frac{1}{c^2} \frac{\partial^2 t}{\partial t^2}[/tex]

    takes the form

    [tex](\nabla^2 + k^2)F(\vec{r},w) = 0 [/tex], which was fairly straight forward.

    I am then told to substitute F(r,w) = G(r,w)e-ikz, rewrite the wave equation in terms of G, and apply the paraxial approximation:

    [tex]2ik\frac{\partial G}{\partial z} >> \frac{\partial^2 G}{\partial z^2}[/tex]

    to get the "paraxial wave equation". So, here's what I have tried...

    [tex]\frac{\partial G}{\partial z} = ikF(\vec{r},w)e^i^k^z = ikG(\vec{r},w)[/tex]

    [tex]\frac{\partial^2 G}{\partial z^2} = -k^2F(\vec{r},w)e^i^k^z = -k^2G(\vec{r},w)[/tex]

    hence [tex]2ik\frac{\partial G}{\partial z} = -2k^2F(\vec{r},w)e^i^k^z = -2k^2G(\vec{r},w)[/tex].

    I also said that [tex]\nabla^2G(\vec{r},w)e^-^i^k^z = -k^2G(\vec{r},w)e^-^i^k^z[/tex] (rewriting my wave equation in terms of G)

    ie. [tex]\nabla^2G(\vec{r},w) = -k^2G(\vec{r},w)[/tex]

    or [tex]\nabla^2G(\vec{r},w) = \frac{\partial^2 G}{\partial z^2}[/tex].

    (so the second derivatives with respect to x and y are zero, don't know if this has any significance?).

    This is essentially as far as I have made it with this question so far. I am not really sure what to do with that so-called paraxial approximation, because when you stick the first and second G derivatives into it and cancel, you get 2>>1, which doesn't really make sense. All I really said beyond this was that

    [tex]2ik\frac{\partial G}{\partial z} >> \nabla^2 G[/tex], and I don't really see how that helps.

    Can anyone offer any suggestions on how to proceed here? I'd really appreciate it.
    Last edited: Nov 28, 2009
  2. jcsd
  3. Nov 28, 2009 #2


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    Homework Helper
    Gold Member

    You need to remember that both [itex]F(\textbf{r},t)[/itex] and [itex]G(\textbf{r},t)[/itex] are functions of the position vector [itex]\textbf{r}[/itex] (and hence [itex]x[/itex], [itex]y[/itex] and [itex]z[/itex])....so when you are taking spatial derivatives of the product of either of these functions with say, [itex]e^{ikz}[/itex], you need to use the product rule.

    For example,

    [tex]\frac{\partial G}{\partial z} =\frac{\partial}{\partial z}\left(F(\textbf{r},\omega)e^{ikz}\right)= ikF(\textbf{r},\omega)e^{ikz}+\frac{\partial G}{\partial z}e^{ikz}\neq ikF(\textbf{r},\omega)e^{ikz}[/tex]
  4. Nov 28, 2009 #3
    No. [itex]G[/itex] is a function [itex]x[/itex], [itex]y[/itex] and [itex]z[/itex], so the second derivatives with respect to [itex]x[/itex] and [itex]y[/itex] should not be zero.
  5. Nov 29, 2009 #4
    Ah of course, can't believe I missed that... thanks guys.
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