Fourier transform of the Helmholtz equation

[Vajhe]

Hi guys, I have been trying to solve the Helmholtz equation with no luck at all; I'm following the procedure found in "Engineering Optics with MATLAB" by Poon and Kim, it goes something like this:

Homework Statement

Homework Equations

Let's start with Helmholtz eq. for the complex amplitude $\psi_p$:

$$\nabla^2 \psi_p + k_0 ^2 \psi_p = 0 , k_0=\frac{w_0}{v}$$

According to the authors, it should be possible to find a solution to that equation applying the two dimensional Fourier Transform to it; just one thing: apparently in engineering, the Fourier Transform is defined like this
$$\int_{- \infty}^{\infty} \int_{-\infty}^{\infty} f(x,y) e^{j k_x x + j k_y y} \, dx \, dy$$

I don't really get why the kernel has positive signs, instead of negatives; the authors mention something to do with a convention of traveling waves, but I don't have the referenced book they mention (Principles of Applied Optics, Banerjee and Poon). If someone could explain me that I will be in a great debt .

Now, my problem is this: according to the authors, the result I should have after some manipulation should be
$$\frac{d^2 \Psi_p}{dz^2}+ k_0 ^2 (1 - \frac{k_x^2}{k_0^2} - \frac{k_y^2}{k_0^2}){\Psi_p} = 0$$

Where $\Psi_p$ is the Fourier Transform of $\psi_p.$

The Attempt at a Solution

When I apply the FT to the Helmholtz eq. I use the Laplace operator in both $\Psi_p$ and $e^{j k_x x + j k_y y}$, that will give me several terms (actually a lot). I do some math and I continue working on it but I don't get the desired expression. I find particularly complicated the total derivative: the Laplace operator gives me partial derivatives, but the result should consist of a total one. Also, several terms are missing. I think a possible solution to the latter problem would be something akin to the relation
$$d = \frac{\partial}{\partial x}dx + \frac{\partial}{\partial y}dy + \frac{\partial}{\partial z}dz$$

But, I don't find an answer about what is $\frac{d^2}{dx^2}$ in partial derivatives (as an operator), the answer I found is not that quite satisfactory, and even in that case, it presents me with some problems like the total derivates of x and y in z (which will be zero in any case).

So I'm pretty stuck at this point, and I would like to see some fresh ideas.

Thanks in advance!

P.D. Sorry if there are some confounding parts, I'm not sure what to do at this point .

 

[strangerep]

Now, my problem is this: according to the authors, the result I should have after some manipulation should be
$$\frac{d^2 \Psi_p}{dz^2}+ k_0 ^2 (1 - \frac{k_x^2}{k_0^2} - \frac{k_y^2}{k_0^2}){\Psi_p} = 0$$
Where $\Psi_p$ is the Fourier Transform of $\psi_p.$

What is $z$ here? Are you sure that's supposed to be there? I would have expected only the 2nd term in the above.

Also, have you checked the standard formulas for the FT of derivatives? (See, e.g., the Wikipedia page on Fourier transforms, under the section "Differentiation".)

 

[Vajhe]

Well, z is the third component of the position. The author explicitly puts it there, but I don't find why.

And actually, I think that the formulas for the FT of derivatives will take me to the answer, but I'm still working on it. I really appreciate the hint (I was very, very stuck).

Let's see if I can fully work the answer in these days and I will put it here for anyone interested in something like this.

Thanks!

 

[strangerep]

Well, z is the third component of the position. The author explicitly puts it there, but I don't find why.

Oh, I see now. It's a 3D problem, but they're only taking a 2D FT. So the $\Psi_p$ is really $\Psi_p(k_x, k_y, z)$.

So presumably, you're supposed to solve for the $z$-dependence first, and then do an inverse 2D FT back to xy space if the final solution needs to be in 3D position space.

 

[blue_leaf77]

It's a 3D problem, but they're only taking a 2D FT

Yes, the author is only taking 2D Fourier transform because for a given wavelength (or frequency), the three components of the wavevector are not all independent - knowing the wavelength, $k_x$, and $k_y$ Is sufficient to compute $k_z$.