# EM Pulse Effects & Calculation: Fourier vs Laplace

• lttung
In summary, the conversation is about electromagnetic pulses and their effects on systems. The question is raised whether Fourier transformation or other types of transformations, such as Laplace, would be more convenient to use in analyzing these pulses. The topic of finding the same energy in different types of pulses is also discussed. One member recommends using FDTD (finite-difference time-domain) for modeling pulses, while another member suggests using finite element or method of moments. The limitations and considerations of using these methods, such as incorporating quantum effects and dealing with inhomogeneities, are also mentioned.

#### lttung

Hi Physics Forum,

I am a student, doing something about electromagnetic pulses.
I want to ask a question:
If we find the effects of EM pulses on some systems, is it convenient to use Fourier transformation to make these pulses look like sinusoidal? Or we can use other kinds of transformations like Laplace one.
Can you recommend me the simplest books telling ways to calculate with EM pulses.

Ah, they tell about the sinusoidal pulse, the Gaussian pulse and the rectangular pulse with the same energy. What does "the same energy" of pulses mean?

Thank you very much.

lttung said:
Ah, they tell about the sinusoidal pulse, the Gaussian pulse and the rectangular pulse with the same energy. What does "the same energy" of pulses mean?

It means that if they hit something and are absorbed, they deliver the same number of joules of energy to the absorbing object.

You could do a Fourier transform, but then you need to solve over a range of frequencies and then do an inverse transform to get the time domain solution. Depending on the type of pulse and the number of points you need to do the inverse transform, this could be more time consuming than a time-domain analysis. It all depends. I reallly simple way to model pulses is to use FDTD, finite-difference time-domain. Allen Taflove is a great authority on FDTD.

Thank two members very much.

It is appealing to learn new method such as FDTD. However, in my problem, the previous author had analytical result of a sinusoidal wave, and there are several surfaces with surface currents (a cell with an organelle inside). I wonder that is this method effective with surface regions, and with the small regions when the quantum effects are important?

I am a student in biophysics, not in engineering fields. So, is this method convenient for soft-condensed matters. I intend to skim such book like "A First Course in Finite Elements". It looks basic and simple for an amateur like me.

Thank you very much.

Quantum effects... I guess that all depends on whether or not you can incorporate them into the differential equations. I have seen FDTD used in quantum problems, the most recent in Casimir force. But that all depends on how you formulate the problem. FDTD is mainly a method to solve partial differential equation, in this case one that is dependent upon time and space. However, FDTD is very useful in inhomogeneous problems. The only problem is that you will have to divide up your inhomogeneities in accordance with a grid. I don't know of a way to use a non-rectangular grid. So if you have a surface with very small feature sizes then you will need to make a small grid which will increase memory and computation time. Using a triangular mesh, like in finite element or method of moments, can result in a more accurate mesh for a given mesh size.

I'm not sure if finite element can model a pulse very well but, I guess it depends on how you do the excitation. I know, though, that I have done a point source current and the results were mediocre. The reason for this is that you assign a current that is "smeared" across a single mesh element. If you have a signal that is supposed to be very small spatially, you will need to make a very dense mesh otherwise the interpolation of the signal across a mesh element may distort it from what is desired. Of course, with a point source current this is impossible to do so there was a bit of deviation.

## 1. What is the difference between Fourier and Laplace transforms?

Fourier transforms and Laplace transforms are both mathematical tools used to analyze signals over a continuous range of frequencies. However, they differ in the types of signals they can analyze. Fourier transforms are used for signals that are continuous and periodic, while Laplace transforms are used for signals that are non-periodic and have a finite duration. In other words, Fourier transforms are best suited for analyzing signals that repeat over time, while Laplace transforms are better for analyzing signals that are one-time events.

## 2. How are EM pulses represented in Fourier and Laplace transforms?

EM pulses are represented in Fourier transforms as a series of sinusoidal waves with different frequencies and amplitudes. In Laplace transforms, EM pulses are represented as a single exponential function with a decay factor that determines the amplitude and duration of the pulse.

## 3. How do Fourier and Laplace transforms calculate the effects of an EM pulse?

Fourier transforms calculate the effects of an EM pulse by breaking down the pulse into its individual frequency components and analyzing them separately. Laplace transforms, on the other hand, integrate the entire pulse over time to determine its overall effect.

## 4. Can both Fourier and Laplace transforms be used to analyze EM pulses in real-world scenarios?

Yes, both Fourier and Laplace transforms can be used to analyze EM pulses in real-world scenarios. However, their applications may vary depending on the type of EM pulse and the desired analysis. For example, Fourier transforms may be more useful for understanding the frequency content of a pulse, while Laplace transforms may be better for predicting the overall impact of the pulse on a system.

## 5. Are there any limitations to using Fourier and Laplace transforms for EM pulse analysis?

One limitation of using Fourier transforms for EM pulse analysis is that it assumes the signal is periodic, which may not always be the case in real-world scenarios. Laplace transforms also have limitations, as they may not accurately represent signals with rapid changes or discontinuities. Additionally, both transforms may require complex mathematical calculations and may not be easily interpretable for non-mathematical audiences.