# Calculating Electric Field Distribution of a Laser Beam

• vivek.iitd
I am using sapphire.In summary, the author is modeling a non-linear phenomenon and they need to calculate electric field distribution of a laser beam. They have done this by taking pulse energy and pulse duration and calculating peak power and then taking beam radius to calculate peak intensity. They then used Gaussian distribution to calculate electric field.

#### vivek.iitd

I am modeling a non-linear phenomenon. For that i need to calculate electric field distribution of a laser beam.

I will be using pulsed laser beam with following parameters:

" Laser pulse with 500-microJ pulses(Ep) at 1.064 micro-m with 12.5-KHz (f) repetition
rate, pulse duration = 20 ns (2*T), and beam radius = 0.2 mm (w) "

From this data i need to calculate, Intensity distribution and from there i need to calculate Electric field distribution both of which are Gaussian in nature. I have done this as follows:

First the peak power = (pulse energy)/Pulse duration = 500μJ/20ns = 25 KW.
Now, taking beam radius to be 200μm, Peak Intensity (Io) = 25 KW/(π*w^2)= 19.89 MW/cm^2;

Now as i know that the distribution is Gaussian, I am using the following equation for creating the distribution.

I = Io*exp(-(t/T)^2)*exp(-(r/w)^2)

Now, I = (cnεo *|E|^2)*0.5

I would like to ask if the above calculation is right.

I thank you for your valuable time.

Are you trying to model the cavity or just the beam ? Electric field distribution of a beam depends on it's mode/polarization so it would be healthier if you could provide more information about your beam.

Did you try to plot your solution as a pattern ?

I am trying to model a cavity, about the mode I am taking it to be TEM00.

Actually i want to see the temporal behavior of OPO (Optical Parametric Oscillator). My approach is as follows:

As i know the temporal profile of the Intensity (Gaussian) I will divide into time slices (each one corresponding to one cavity round trip). For one time slice i will take intensity to be constant and from there i will calculate Electric field distribution (in spatial x-y coordinates, it is also Gaussian) and will propagate it back and forth in the cavity (z-direction). I will continue the process for one laser pulse.

Please let me know which other parameters you will need?

You are trying to see reverse SHG effect, for this you need to first model the crystal by defining it as an anisotropic medium with necessary dispersion parameters (μ,σ,ε).

For simplicity, you can model OPO as a blackbox with one input and two outputs. If you don't want to go into trouble to model the crystal, you can model it as a beam splitter with different wavelength outputs.

Modelling the cavity is not that simple unfortunately. You need to model both mirrors and the crystal for complete behavior of cavity. But this might only be necessary to a precision point, it ends up in your error tolerance.

TEM mode beam will not have any electric field component along the propagation path. You can plot Gaussian pattern of your beam but that will not include electric field.

vivek.iitd said:

I am trying to model a cavity, about the mode I am taking it to be TEM00.

Actually i want to see the temporal behavior of OPO (Optical Parametric Oscillator). My approach is as follows:

As i know the temporal profile of the Intensity (Gaussian) I will divide into time slices (each one corresponding to one cavity round trip). For one time slice i will take intensity to be constant and from there i will calculate Electric field distribution (in spatial x-y coordinates, it is also Gaussian) and will propagate it back and forth in the cavity (z-direction). I will continue the process for one laser pulse.

Please let me know which other parameters you will need?

Regarding dispersion parameters, as i will be using Quasi Phase Matching condition, it will simplify my problem. Of course the TEM mode will not have electric vector along the propagation direction, it will be in transverse plane (perpendicular to the direction of propagation).

I can model the crystal, as i have necessary coupled differential equations with me. Mirrors can be modeled using ABCD matrix. I certainly agree with you, it is not that simple...

Right now i am using the following strategy to model the whole phenomenon. I will have the intensity profile of the laser beam from there i can calculate the electric field profile in the transverse direction and i can propagate it inside the cavity by solving coupled differential equations while the beams are inside the crystal, then i will reflect the beams at the right mirror, thereafter i can propagate the beam towards the left mirror and and when the beam will reflect back it will give me one round trip.

The input beams will be pump (from the laser), signal and idler (vacuum fields at the starting).

Sounds like you are on the right track. Are you constructing the intensity profile from analytical formulation or are you using a macro-like profile ? Just keep in mind that the behavior you will calculate inside the crystal will probably be different than the real-world behavior inside.

Which crystal are you using and which platform you use to simulate ?

vivek.iitd said:
Right now i am using the following strategy to model the whole phenomenon. I will have the intensity profile of the laser beam from there i can calculate the electric field profile in the transverse direction and i can propagate it inside the cavity by solving coupled differential equations while the beams are inside the crystal, then i will reflect the beams at the right mirror, thereafter i can propagate the beam towards the left mirror and and when the beam will reflect back it will give me one round trip.

The input beams will be pump (from the laser), signal and idler (vacuum fields at the starting).

I am using KTP, and i am doing my simulations on MATLAB. Regarding Intensity profile I am just using the following Gaussian distribution to calculate Intensity at different radial points.

I(r) = Io*exp(-(r/w)^2), from where i am calculating electric field distribution.

Also solving, coupled differential equations with electric field components having Gaussian distribution in the transverse plane is also numerically challenging because the amplitude will vary for different r. In case of plane wave the amplitude is constant in transverse direction so it is much simpler.
I just have to run the simulations and see whether it gives me correct results...:|

KTP is an easy one, you can model it precisely without too much of an effort.

Plane wave assumption will definitely shrink the size of the matrices. Just keep in mind that you assumed quasi phase matching case for dispersion parameters and now you will assume the plane wave instead of time-varying field components. Each assumption will get you far from the real world results.

Run the sim and let's see what happens.

vivek.iitd said:
I am using KTP, and i am doing my simulations on MATLAB. Regarding Intensity profile I am just using the following Gaussian distribution to calculate Intensity at different radial points.

I(r) = Io*exp(-(r/w)^2), from where i am calculating electric field distribution.

Also solving, coupled differential equations with electric field components having Gaussian distribution in the transverse plane is also numerically challenging because the amplitude will vary for different r. In case of plane wave the amplitude is constant in transverse direction so it is much simpler.
I just have to run the simulations and see whether it gives me correct results...:|

## 1. What is the formula for calculating the electric field distribution of a laser beam?

The formula for calculating the electric field distribution of a laser beam is E(x,y,z) = E0(x,y,z) * exp(-i(kz-wt)), where E0(x,y,z) is the initial electric field at the source, k is the wavevector, z is the propagation direction, and t is time.

## 2. How do you determine the wavevector and propagation direction in the formula?

The wavevector, k, is determined by the wavelength of the laser, which can be measured using a spectrophotometer. The propagation direction, z, is the direction in which the laser beam is traveling.

## 3. Can you calculate the electric field distribution for a laser beam with different wavelengths?

Yes, the formula for calculating the electric field distribution is valid for any wavelength of laser. However, the wavevector, k, will change depending on the wavelength.

## 4. Are there any assumptions made when using the formula for calculating electric field distribution?

Yes, the formula assumes that the laser beam is monochromatic (single wavelength) and that the beam is traveling in a uniform medium with no obstructions or reflections.

## 5. How is the electric field distribution graphically represented?

The electric field distribution can be graphically represented using a 2D or 3D plot, with the electric field intensity plotted against the spatial coordinates of x, y, and z. This can help visualize the distribution of the electric field along the laser beam's path.