# A problem on the paraxial wave equation?

• jeebs
In summary, the author is trying to solve a problem involving the paraxial approximation for a scalar wave, and is not getting very far. The first part of the question is proving that the homogeneous wave equation takes the form \frac{\partial^2 G}{\partial z^2} = -k^2F(\vec{r},w)e^i^k^z. However, when he tries to substitute in G(r,w) using the paraxial approximation, he gets an equation that is not solvable. He is then told to use the product rule to solve for G(r,w), but this still does not yield a solvable equation. He suggests that perhaps the
jeebs
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(r,w)eiwt.

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

$$\nabla^2 \Psi = \frac{1}{c^2} \frac{\partial^2 t}{\partial t^2}$$

takes the form

$$(\nabla^2 + k^2)F(\vec{r},w) = 0$$, 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:

$$2ik\frac{\partial G}{\partial z} >> \frac{\partial^2 G}{\partial z^2}$$

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

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

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

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

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

ie. $$\nabla^2G(\vec{r},w) = -k^2G(\vec{r},w)$$

or $$\nabla^2G(\vec{r},w) = \frac{\partial^2 G}{\partial z^2}$$.

(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

$$2ik\frac{\partial G}{\partial z} >> \nabla^2 G$$, 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.

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

For example,

$$\frac{\partial G}{\partial z} =\frac{\partial}{\partial z}\left(F(\textbf{r},\omega)e^{ikz}\right)= ikF(\textbf{r},\omega)e^{ikz}+\frac{\partial G}{\partial z}e^{ikz}\neq ikF(\textbf{r},\omega)e^{ikz}$$

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

No. $G$ is a function $x$, $y$ and $z$, so the second derivatives with respect to $x$ and $y$ should not be zero.

Ah of course, can't believe I missed that... thanks guys.

## 1. What is the paraxial wave equation?

The paraxial wave equation is a mathematical equation used to describe the propagation of electromagnetic waves in a medium, where the angle of the waves is small and can be approximated by a parabola.

## 2. What are the applications of the paraxial wave equation?

The paraxial wave equation is used in various fields of science and engineering, such as optics, acoustics, and electromagnetics. It is used to model and analyze the behavior of light, sound, and other electromagnetic waves in different mediums.

## 3. How is the paraxial wave equation derived?

The paraxial wave equation is derived from Maxwell's equations, which describe the fundamental laws of electromagnetism. It is obtained by assuming that the angle of the wave is small, and using a series expansion to simplify the equations.

## 4. What are the limitations of the paraxial wave equation?

The paraxial wave equation is only valid for small angles of wave propagation and for homogeneous mediums. It also does not take into account effects such as diffraction and scattering, which can significantly affect the behavior of waves.

## 5. How is the paraxial wave equation solved?

The paraxial wave equation can be solved using various methods, such as analytical solutions, numerical methods, or computer simulations. The specific approach used depends on the complexity of the problem and the desired level of accuracy.

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