Measuring the beam in two cross-sections along the propagation axis would be useful to avoid artifacts. If you were using a Gaussian equation and managed to fit the curve to your experimental data, that means that your beam is a Gaussian beam with the accuracy that can be estimated from the residual error.

Hi ! If you want to know if it a Gaussian beam propagation, you should measure the beam at various distance from the waist (location where the beam is the smallest) with a camera and extract the 1/e^2 diameter en the X and Y axis. Then you can fit the curve with the Gaussian function given on my picture. Axis on the picture are distance from the laser beam (cm) and beam diameter at 1/e^2. More information there : https://www.rp-photonics.com/gaussian_beams.html

Thank you so much Leonid Vesselov for your answer. Just for the justification, let us consider I have recorded the 2-d intensity profile and I do not have any idea whether it is Gaussian or not. Then how to check it is a Gaussian distribution. As mentioned by Loïc Meignien , we can check the laser beam is the Gaussian beam by taking the measurement with varying the distance. However, it is very necessary to know distribution is Gaussian at any location.

Yet another method to check the normal distribution using the Full width at Half maximum. For a normal distribution . FWHM = 2*\sigma*\sqrt(2ln2). Where sigma is the standard deviation from the mean. Then for the normal distribution the value of sigma should be equal to 0.425 times FWHM. Here we can find out the sigma using the Gaussian fitting curve and FWHM from the linear plot from the data. unfortunately this is necessary condition but not sufficient. On the basis of this link I am trying to explain (https://mathworld.wolfram.com/GaussianFunction.html). However it will be very useful to find out any other method to check the data is Gaussian distributed or not. Thank you so much. Please let me know you have any other lead.

If the beam shape is well fitted with a Gaussian function at least at two distances along the propagation that means the beam is Gaussian and its parameters can be calculated and predicted from Gaussian beam equation. See wikipedia for equations.

Nagendra Kumar ：

1. you have the measured normalized intensity data I(x,y,z), from which you can assume the electric field as E2(x,y,z);

2. you assum a ideal normalized Gaussian intensity distribution G(x,y,z), which is a function of assuemd wasit of w, you get electric field E1(x,y,z);

3. instead of ask the question of how well it is from a ideal Gassian beam, we ask the question: how much engergy of the distribution I(x,y,z) can match with (coupling to ) a pecfect Gaussain Beam G(x,y,z), we can use following equation to calculate the efficicency of coupling:

Continue:

1. Above equation need a complex distribution of measured electric field, however you don't have the phase information in most case , above equation is only correct at the waist (usually at the z-location of smallest beam size), that means you should use the bottom data from the figure provided by previous answer from Loïc Meignien;

2. different ideal gaussian beam have different coupling efficiency, vary the waist in the function of G(x,y), the maximum efficiency means the best fitted gaussian beam;

3. this method is only correct if the phase variation at waist is zero or very samll， otherwise you need measure the phase as well together with intensity, which is non-impracticable.

I don't really understand you question : if you want to know is the transverse distribution is Gaussian, you need a beam profiler to measure the intensity profile. Most of them have an automatic fit. If you want to know is the propagation of the beam is Gaussian, you need the same beam profiler and make some measurement along the Z axis as I mentioned previously. If the fit along the Z axis is perfect this means that the phase is zero at the waist : perfect solution of the propagation equation => TEM00 mode. If you want to know it in only one measurement at one location you need the phase : only a phase sensor like Phasics SID4 or Imagine Optics HASO can do it and thus give you the M squared of the beam.

Dear Loïc Meignien,

Thank you for your recommendation. I partially agree on the functionality of M squre parameter.

I don't know whether Nagendra Kumar has phase measure instrumment like Phasics SID4 on hand, if yes it would be perfect, then he can get a complex electrical distribution of E2, which provide the accurate intergral from equation of above figure.

Yes, The beam intensity measurement instrument can provide gaussian fitting and it is a good fitting for a perfect gaussian beam, the reason why Nagendra Kumar ask the question, to my understanding , is about how well the fitting is in the case of non-perfect gaussian beam measurement (Nagendra Kumar:pls correct me if I misunderstood your question), M square can be a parameter, but I think overlap integral provided more information because it tell us how much enengy can be fitted into the ideal gaussian beam.

I also thank you for providing the information about phase measurement instrument like Phasics SID4 which makes phase measurement practicable.

I would like to thank Bo Chen for your detail explanation regarding my question. I would like to let you know, I have a basic question regarding to test the Gaussian distribution and it may be a silly kind question. 

Since,  I am working on an experiment that uses a laser source and combination lens and camera. First we collimate a diverging beam emerging from the tip of an optical fiber using a lens of 100 mm focal length then I crop it using iris diaphragm of 8 mm diameter. We capture the images of a beam of light that is brighter in the center and dimmer towards the edges. Theoretically, the intensity profile of this beam should be Gaussian, and by eye, it looks like it is. In fact, if we least-squares fit a Gaussian curve to it, it looks like a very reasonable fit, but we would like to know if there is a statistical test that can give us an idea of the confidence with which we can claim that the profile is indeed Gaussian.

I hope I am able to explain my question.

I really appreciate the all the valuable response to question.

It is clear, that a Gaussian beam is a mathematical model. Measurements results of the beam at several distances along the propagation axis will give you an idea of how close the real beam to the Gaussian model. The measure could be M2 parameter or residual errors if measurement data fit to the model. there are plenty of methods of the beam profile measurement available. What else are you looking for???

On my opinion the simplest way to solve the task it is using the diffuser - fine-grained thin matte film. Spot of a laser beam is projected on the film and, then by help lens the image of this spot, with the required magnification, is projected onto the CCD matrix.