There are several ways to do that, depending on the assumption, if you consider a plane wave being focused, in scalar mode (simplest case), to a Gaussian laser beam, polarized, and taking into account the immersion medium, the cover, spherical aberration into the sample etc… You may start from Born and Wolf's approach (chap 6 if I remember well), 3-D focusing. You should also precisely define what the spot size is, depending on your application, it may slightly vary… FWHM, FW at 90% etc.. There are some microscopy (deconvolution) softwares that may help : COSMOS, ImageTrak have modules to compute PSFs. Have a look ! Hope it may help.

I will look into that and come back if my doubt still persists

Why not image calculating? Capture real time pictures, calibrate and calculate?

I hope it helps you.

Why do you state that "Rayleigh criteria doesn't help in this case"?

sorry for the stupid counterquestion?

Samuel Arba Mosquera: If you use rayleigh criteria, we violate the assumption that it holds good for uniform illumination. Uniform i mean, intensity. Second the spot diameter turns out to be lesser than the wavelength of the radiation. Which is again violating the theoritical limit of diffraction limited spot size.

@xizhang chen : I need theoritical estimation of the spot size and then it can be cross checked using the image.

So why don't you use the expression (as per Rayleigh criterion):

spot_size = 1,22 * lambda / NAeff

to get spot size smaller than lambda, you need NAeff larger than 1,22, which is rather high and unlikely.

Further you may either use other factors instead of 1,22 if other cross-sections different from uniform illumination (e.g. 1,22*2^0,5 ~1,73) for Gaussian beams.

Equally you may calculate NAeff considering the local contributions of the different energy proportions of "consecutive" NA rings, ie NAeff = NAnominal for uniform illumination but NAeff ~0,87NAnominal for gaussian illumination

Combining both you get that the spot size of a Gaussian beam is approximately:

spot_size = 2 * lambda / NAnominal

For beam profiles between gaussian and flat-top or uniform illumination, you get spot-sizes in between both limits

but I may be wrong also with some assumptions here...

Dear Karthikeyan, you can see, that it is difficult to recommend resonable solution of this simple task, if you do not explain us your conditions more detail.

May be you should take a lens with large focal length F2, determine the focal spot size D here ,according your criteria, after that calculate spot size d for microscope lens F1: d =D x F1/F2;

I did it with a pixel detector of known size. Well you can also use the imaging (using a mobile camera of an outdated mobile which you need to spare for the same)

first of all you have knoe, - do you have gaussian, flat-top or other intensity profile. If the laser in in fundamental mode, i.d TEM00 - use the transformation of gaussian beams (to see Wiki- pedia).

Remember: the Gauss profile (exp(-x2/a2) always remain Gauss, also in focus of lens. Instead the flat top profile have sinx/x distribution in focus.

our's is an gaussian beam with TEM00 mode..

the aim of focusing is to establish the position and the waist value. To see an calculation example http://www.rp-photonics.com/gaussian_beams.html

Calculate using wo=2*f*wavelength/(Pi*beam diameter on lens) [ chapter 3.2, Fundamentals of Photonics, by Saleh]. It is also possible to measure the spot size (eg., knife edge method ) at few points along the beam near the focus and fit the data(z and Wz) to the equation describing the relation between spot size and distance (Wz=wo[1+M2*(z-zo)^2/(zr^2)]^0.5) where M2 is M^2 value of the beam, zo is the location of waist, wo is waist radius which are parameters to be found by the fit . zr is Rayleigh range.

Which laser? What is the wavelength? A practical way : you may measure the spot size practically then correlate with the formalisms given above and calibrate. Remind that phase front may vary the spot size even the beam profile is TEMoo.

@Srikanth gurram... I thank you for giving your valuable answer. But the formula you mentioned gives spot less than its diffracted limited spot, which is not expected. We are trying to calculate the spot size of laser focussed by 100X microscope objective. We cannot do that knife edge experiment.

@ Parviz Parvin... We want to calculate and then compare with the measured spot size.

The following formalism gives the simplest estimation:

Teta (diffraction Limit)=1.22 lambda/D

r= Teta * f

A=Pi* r 2

Where lambda, D , r, f and A denote to be laser wavelength, laser diameter, radius of focused area( spot size), focal length and focal area. however, the success depends of the accuracy of the measurement and how the beam profile is close to TEMoo Gaussian profile ( M2 factor ~ 1.1). Assuming the no lens aberration !

You can take a snapshot of reflection intensity of the laser through the 100x objective. And, fitting of intensity profile. imageJ can be used for this, it has curve fitting feature (i.e. gaussian)

I followed the procedure what Jae-Hyuch noted but using a pixel camera (modified pixel camera, known pixel size-from company as well as estimated), a spot image can be taken. By taking a line profile from the image and Gaussian fitting using FREE IMAGEJ we can get the full with half maxima.

Have you found a way to calculate the spot size? what was the measured spot size?

Thank you.

beam waist radius=2*lambda*focal length/pi/beam diameter

Sanjoy Paul:

The scale of *the reflected image* should be calibrated using known scale before the fitting. CCD pixel size can be directly used when the CCD is located in focused region, which is not an easy task sometimes and the 100x objective makes small beam that is smaller than typical CCD pixel size (~4 um).

Have you thought about the D-square extrapolating method? It is fairly down-to-earth simple and practical...

http://buildlog.net/cnc_laser/laser_calcs.htm

dear Parviz Parvin;

In your presented equation, how can i calculate the D (laser diameter) and f ( focal length)?

sincerely yours

First consult the following tutorial

https://www.newport.com/n/gaussian-beam-optics

Now that you have become famI liar with Gaussian-beam-optics beams you may try the following on line calculator: https://www.controllaser.com/services/buyers-guide/spot-size-and-depth-of-focus-calculator/

A more elaborate on line calculator follows:

https://www.ophiropt.com/laser--measurement/laser-power-energy-meters/services/focal-spot-size-calculator-for-gaussian-beams

You would may try to do the Knife Edge Technique to measure the actual focused beam size of the laser beam if your beam profile is closer to the guassian beam.

W=wavelength*f/bi*w0

w0= beam waist

Using a carbon paper after the laser beam falls on it

by CCD Camera

https://www.ophiropt.com/laser--measurement/laser-power-energy-meters/services/focal-spot-size-calculator-for-gaussian-beams

Those are all about theoretical value of a laser probe beam. In real measurements, when laser irradiated on the flat surface of powder materials, its beam size would be expanded and much higher than its theoretical value.

https://www.lasercalculator.com/laser-spot-size-calculator/

Hi Dhanapal,

The correct way for the precise measurement of the laser spot is using beam profiler. It can give you the value in X and Y with an error bar of few nm.

It would be easier to use Beam profiler to a beam spot on a flat surface of, e.g., glass, wafer and metal. However, it would be more difficult or impossible to get it done on a surface of pressed powder, accurately or correctly. In addition, sometimes, signals of some materials outside the beam spot can also be detected due to reflection/refraction/deflection/scattering, you name it, which make it more difficult to obtain a right beam spot size.

Or sampling size, including sampling area (surface) or volume (plane area by depth) respectively, would have more practical values than theoretical beam size in real measurements.