Solving Wave Equation with Paraxial Approximation

[Confundo]

Homework Statement

Homework Equations

http://books.google.co.uk/books?id=...hl=en&sa=X&oi=book_result&resnum=10&ct=result

The Attempt at a Solution

I don't understand how they're gone from (4) to (5) after making that substitution for U_o the $$k^2U_0$$ term becomes a derivative why?

 

[gabbagabbahey]

If $U_0(r,z)=V(r,z)e^{ikz}$, then:

$$\frac{\partial U_0}{\partial z}=\frac{\partial V}{\partial z}e^{ikz}+ikVe^{ikz}$$

due to the product rule.

so... $$\frac{\partial^2 U_0}{\partial z^2}=\ldots$$?

 

[Confundo]

Oops, I see now I was looking past the obvious. I'll do the calculation in the morning. Is there a reason why they leave out the $$e^{ikz}$$ from the final equation?

Thanks.

edit: $$V = V(r,z)e^{ikz}$$?

 

[gabbagabbahey]

Oops, I see now I was looking past the obvious. I'll do the calculation in the morning. Is there a reason why they leave out the $$e^{ikz}$$ from the final equation?

They haven't "left it out"...To see what becomes of that term, work out the derivatives in equation (4) using the assumed form of U_0.

edit: $$V = V(r,z)e^{ikz}$$?

Huh?! ...What are you trying to ask here?

 

[Confundo]

Worked that out now, must learn not to try and do derivatives in my head while tired.