The beam waist (w0), or smallest spot size, for a Gaussian beam after a lens is given by 

2w0 = 4 lambda / pi * f/d

where f is the focal length of the lens and d is the diameter of the beam prior to the lens.

In terms of experimental measurement/verification, you could use a beam profiler on a linear translation stage to measure the beam diameter at different distances along its beam.

If you don't have a beam profiler, you might be able to get away using a camera, but another method would be to use an intensity meter and a knife edge to measure the beam profile. 

Answers to your questions depend on what you are aiming at. Alexander has given you the formular for best focus of far field in free space. This is not related to lasers though, and was obtained and experimentally verified before lasers were invented. I remember measurements of chromatic aberrations of a mirror for different intensity distributions in the beam (sharp borders or gradually increase of absorption).

If you are not limited by far field, then look also at the SNOM (scanning near field optical microscopy) technology. By the way, it is used also to profile the waist of a focus.

Dear,

After going through a lens, the beam becomes a hyperboloid. At the point where the density of the light is most intense, the beam does not pass through a focal point but a focal spot (See attached file).

Z0=(2×λ)/π √(p^2-1)×((2×f)/W)^2

W0=(2×λ)/(π×θ0)     with       θ0~W/(2×f)

Z0 : Depth of field in mm

f : Distance of the focal point in mm

W0 : Diameter of the focal spot in mm

λ : Wavelength in mm

W : Diameter of the beam before focusing in mm

p : Tolerance factor

Thank you all! My aim is obtain a very very small focal spot (the smaller the better). But  I don't know if it is limited by some principles such as diffraction limit. And is there any experimental proof for the possible limit?

The size of the spot is limited by diffraction.  Alexander Malm's response gives the spot diameter for a Gaussian beam, but smaller diameters are possible if the aperture of the lens is illuminated more uniformly - for example with a Gaussian beam whose diameter is much larger than the lens aperture.

For a uniformly illuminated lens aperture, the intensity distribution is described by the Airy disk https://en.wikipedia.org/wiki/Airy_disk.  The radius of the first minimum is given by: 1.22 Wavelength / Numerical_Aperture - approximately equal to the diameter at half maximum intensity.

Spot size is minimised using a high numerical aperture lens.  For a lens in air, the maximum theoretical numerical aperture is 1.0.  Higher numerical apertures are possible if the space between lens and subject is filled with a higher refractive index medium.  Oil immersion objectives are manufactured with numerical apertures of 1.3 or 1.4. http://www.olympusmicro.com/primer/anatomy/numaperture.html

If your first aim is to obtain the smallest spot you can achieve, you may consider about changing your laser wavelength.. The shorter the wavelength, the smaller the beam waist.. You can see it from every equations described above and is very well known when you deal with optical beams.

Dear Miguel Llera,

You are right. Decreasing the laser wavelength is really a choice. However, the spot I have now is much larger than the wavelength, so I will decrease the spot firstly by the focusing system and then change the wavelength if necessary.

Thanks for all of help!!

You can also get smallest spot size by using very small focal length lens or objective or aspheric lens without changing the wavelength. Also, you can expand the initial beam size (d) that also reduced the spot size after focus.

Qiuyu Xia, if you really want just a small spot (not the highest power density), then using a SNOM tip is a simple way to go. It will allow you to have a spot of 50 nm or even smaller size.

For highest power density, the most important is to take care about f/d ratio (f/W in the other formula). Next important is divergence of the laser beam, optics quality, and finally, intensity distribution in the beam.

Vladimir I Chukharev, thanks for your advice. I am looking up the SNOM, and it may take a period of time to understand why it breaks the diffraction limit.

The spot size is limited by three components:

1) The diffraction limit given by the ratio of wavelength to the beam diameter (not lens diameter)

2) The divergence of the laser beam. Usually around a mrad of non parallelism which cannot be removed by optical means can be assumed.

3) The spherical aberration of the lens used. This term enters with the third power of the ratio of the beam diameter to the focal length.

So you have to take a choice or a compromise. One idea is to artificially widen the beam before re-focusing, but this shortens the beam waist and increases the angle of illumination of a target. And it increases the term of spherical aberration quite a lot.

The latter can be minimized using aspherical optics or using a achromatic lens. This sounds a little bit strange with respect to the monochromatic beam, but these lenses are also corrected for spherical aberration. There are also "laser monochromatic" lenses optimized for that. Mounting simple lenses the correct way also helps a lot :-). For pulsed lasers mind using air spaced lens systems.

@Rolf Bomback: much of what you say is valid, but seems to assume a fixed lens focal length, which to my mind is an unnecessary constraint.

Rather than just the wavelength and the beam diameter, the diffraction-limited spot size depends on the wavelength and far-field distribution of light generated by the lens at the focus. The angular range is conveniently specified via the Numerical Aperture, the sine of the half-angle of the cone of rays incident at the focus, multiplied by the refractive index of the medium (unity for air or vacuum).

Alexander Malm's reply gave the formula for a Gaussian beam focused to a Gaussian spot. The beam waist depends on the ratio of wavelength to beam diameter multiplied by the focal length of the lens.

In practice the beam profile at the lens will be vignetted by the lens entrance pupil. My earlier reply gave the formula for the Airy disk radius when the lens entrance pupil is uniformly illuminated.

More generally, both incident beam profile and lens numerical aperture must be considered. Truncating the Gaussian beam profile results in a focus spot which is larger than from either an extended Gaussian profile, or from the same lens with a uniformly filled lens aperture - as shown in the figure below. (EDIT 2016-11-03).

As Souvik Sil proposed, if the beam does not fill the lens aperture, an alternative to widening the beam is a shorter focal length lens with a high numerical aperture such as a microscope objective or aspheric lens.  Using a short focal length minimizes the impact of spherical aberration.

Of course, truncating a Gaussian beam inevitably loses some of the energy.  If smallest spot size rather than coupling efficiency is paramount, then Vladimir I Chukharev's suggestion to use a scanning near field probe tip will be more effective than a lens coupling scheme.

@Alan Robinson: I don't know if I have misunderstood your description to that sentence

"Truncating the Gaussian beam profile results in a focus spot which is smaller than from either an extended Gaussian profile, or from the same lens with a uniformly filled lens aperture - as shown in the figure below."

In the left part of the figure, does the truncating Gaussian beam profile correspond to the Red line? And in the left part of the figure, the spot of the red line is bigger than the others. This may violate to your description.

@Qiuya XIa:  Correct - truncating the Gaussian beam at the lens aperture results in a wider spot - as expected from diffraction.

Thanks for pointing this out.  I have corrected my original reply.  

Alan.

 @Alan Robinson: very well explained!

Rolf wrote " So you have to take a choice or a compromise. One idea is to artificially widen the beam before re-focusing, but this shortens the beam waist and increases the angle of illumination of a target."

With a fundamental Gaussian beam the phase front is plane at the focal plane, so there is no variation of angle of illumination in the focal plane.  However, the wavelength at the focal plane is longer for small waists, and there there will be a big variation in wavelength and direction just before and after the focal plane, because there is 180 degrees phase slip on axis that has to take place through the waist in a distance that is short when the waist is small.  I guess it depends on what depth of focus you require.

I think that the limit to waist size is about 2wo = half a wavelength.  Any smaller than this requires non-propagating (evanescent) modes that will attenuate too much before they get to the focal plane unless you can start them with very high amplitude or generate them very close in - negative refractive index materials do this.

Mount a small hole at the focal plane, then you might get faked small focus spot. Otherwise, the focal spot size is always larger than the wavelength.

despite any limiting factors, could the Gaussian beam be focused down to the wave length scale? and how?

Mohammed A. Saleh: what do you mean by "focused down to the wavelength scale"?

If you use a high numerical aperture lens to focus the beam, the spot size depends on the intensity profile of the beam at the lens principal plane, and the extent to which this is truncated by the lens entrance pupil. This is illustrated in my 2 November 2016 answer to this question.

If the Gaussian beam is much wider than the entrance pupil, the diameter of the first minimum of the Airy disk diffraction pattern is 1.22 λ / NA. For wavelength λ 500 nm, and lens numerical aperture 0.9, this is 1.36 λ = 678 nm. The spot diameter at half maximum intensity (fwhm) is 286 nm, and 457 nm at 1/e2.

If the Gaussian beam 1/e2 diameter is matched to the lens entrance pupil, the spot diameter increases slightly to 314 nm fwhm. The profile is not precisely Gaussian, but the diameter at 1/e2 intensity is 508 nm.

Note that objective lenses are manufactured with numerical apertures as high as 0.95 in air, or 1.45 if an oil immersion lens is acceptable.

https://www.olympus-lifescience.com/en/objectives/uplfln/

https://www.olympus-ims.com/en/microscope/mplapon-oil/

focused down to the wavelength scale= I mean that the minimum spot size equal to laser wavelength.. tanks for your answer

Agreed with

@ Mohammed A. Saleh

请问，如果入射的不是平行光（没有准直，是发散的），那么他出射的最小光斑是多大？

Hi Shenghang Zhai , if you post your answer in English, many people can read it. Request you to post a copy in English too