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jeebs

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Hi,

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 traveling in the z-direction, and says that a scalar wave has the form F(

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(

[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.

Thanks.

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 traveling in the z-direction, and says that a scalar wave has the form F(

**r**,w)e^{iwt}.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.

Thanks.

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