The size of the laser spot does not depend on the duration of the laser pulse. With the change of the pulse will change the intensity (power) of the laser radiation. With given focusing optics and laser energy spot size is determined by the divergence of the laser radiation. To solve this problem is better to use multiple synchronized lasers with different divergences whose radiation coaxially. But it is a difficult task.

The size of the laser spot does not depend on the duration of the laser pulse. With the change of the pulse will change the intensity (power) of the laser radiation. With given focusing optics and laser energy spot size is determined by the divergence of the laser radiation. To solve this problem is better to use multiple synchronized lasers with different divergences whose radiation coaxially. But it is a difficult task.

Dear Leonid

Thanks a lot for your responce. 

We have thought about using deformable mirror or liquid crystal spatial light modulator (LCSLM) to modulate the wavefront of the light. As a result, the focal spot can be changed. But the shortest  responce time of piezoelectric materials or liquid crystal is about several ms.  So if the pulse width is larger than 10ms, we can use these two methods. But our pulse is so short that they can not responce in such time. 

Electro-optic modulator has enough fast responce, but it's not spatial device, witch can only modulate the temporal phase of the pulse, not spatial phase. Maybe we can add gratings after the phase modulator, which can translate the temporal modulation to spatial modulation, but it's very complex to design and make suitable gratings.

The method using two coaxial beams you mentioned is a good idea. Maybe it's the only feasible way. Thank you!

If you have a polarized laser beam, you could make a lens from a birefringent material and then modulate the polarization of the pulse during the pulse using a Pockels Cell for example. This will shift the waist position so you could position your experiment relative to the waist to exploit the shift? Alternatively use the change in polarization to change optical paths and incorporate a second lens in one of the paths to change the focal properties?

It is well known that in the case of mutually orthogonal polarizations of said waves ( for example, mutually perpendicular linear polarizations , or right and left circular ) above the spatially periodic intensity modulation does not occur. On this basis, often made erroneous conclusion, that waves with mutually orthogonal polarizations do not interfere at all. In fact, the interference occurs only in contrast to the co-polar space- periodic modulation (in this case is not subject to the intensity of radiation and its polarization state). Depending on the phase difference between the resulting polarization will change from linear to elliptical (left or right). Therefore, the spot shape will also change. You are satisfied with it? Moreover, the electro-optic element through the Pockels effect requires a significant amount of control for directing the stresses (> 1 kV), and it laser strength can limit of the application area.

If a very quick change of the focal position, somewhere on a sine-function, is good enough for you, you might want to look for details of a talk I just heard two weeks ago:

High-speed variable optical elements for advanced laser manufacturing

Prof. C.B. Arnold

Mechanical and Aerospace Engineering, Princeton University

Position or dimension?

Focus position on beam axis, therefor the dimension will chance.

Dear Alexander,

Is the conference you attended LANE2014? I can't find similar papers titled like the talk, but only these two:

M. Duocastella and C. B. Arnold, “Enhanced depth of field laser processing using an ultra-high-speed axial scanner”, Appl. Phys. Lett., 102, 061113 (2013)

M. Duocastella and C. B. Arnold, “Transient response in ultra-high speed liquid lenses”, J. Phys. D: Appl. Phys., 46, 075102-09 (2013)

Did I find the right papers?

Dear John,

Thanks. It's a new idea to modulate the polarization of the pulse.

But as Leonid said, light transmit in birefringent material has only two parts: ordinary light and extraordinary  light, the refractive index of o light is always 'no' and the othor one is 'ne'. So, if the birefringent material is fixed, the spot  has also two parts and their intensities can changed as the polarrization change, but their shapes are fixed.

So there are only two states of the spot we can control: one is all o light, one is all e light. Two states is better than one, but it's difficult to make a high-precision lens with birefringent crystal because crystal is too soft to process, especially large aperture lens.

Thanks all the same for your responce and new idea.

Hi Yuanchao: I may have missed something about the argument against the birefringent lens. If I take a birefringent material with two orthogonal axes with n-o and n-e and make say a plano-convex lens from it, the lens-makers formula relates the focal length of the lens to the radius of curvature of the lens and the index of the material. If I send in a beam of light with a pure linear polarization state aligned along n-o for example I get a focal length related to the radius and n-o? If I then rotate the polarization through 90 degrees to be linear but aligned with n-e, do I not get a different focal length, the difference being proportional to the difference in index? There are several materials with birefringence that are hard and would make good lenses. Sapphire, lithium niobate, rutile, YALO, (Vanadate) etc.

Thinking about this more I think that two bigger problems with the birefringent lens idea are that the spot may not be round at the waist - you may have a circle of least confusion, and that the focal point will displace laterally in X/Y in addition to the waist position shifting in Z because of walk-off. 

There aren't many mechanisms that will affect a beam in 10 ns. Path switching using polarization and a Pockels cell then using polarization recombination will allow you to switch the beam into a path where you can add mild focal power to move the position of the focus. You can do the recombination in a way that keeps the spots overlapped in X and Y. The downside, if this is an issue for you, is that the polarization will be flipped during the pulse, however if you are not sensitive to helicity, you could add a quarter wave plate after the recombination to convert the two linear polarizations in to left and right circular. If you have a really large beam profile the cost of the Pockels cell could get prohibitive. At 1064 nm an 8 mm clear aperture with excellent uniformity is about $3k.

I don't think these are right. I would try to contact Prof. Arnold and ask about his talk on the Lane2014. You can  also find some information on his homepage:

http://www.princeton.edu/mae/people/faculty/craigarnold/

Dear John,

It's so kind of you to share your rich experience and knowledge with me. Thank you for your advice on choosing crystal and analysis on how to use the polarization modulation of pulse . I'll consider what you said.

Dear Alexander, thanks for the help. I'm waiting for your good news. 

Are you sure that you need to modulate the wavefront of a laser pulse? Or that you need real time actuator, as opposed to a predefined sequence to be performed?

Because by changing over time the width of the beam (diffraction limit), its divergence, its color (chromatic dependence of the diffraction limit, chromatic aberration of the focusing optics) or a by number of other ways you can program the beam to change its final size. Is it broadband? Can you chirp the beam and shape its spectrum?

More data is useful for framing the application: type of source, wavelength, power, polarization sensitivity... What are the actual constraints? How rigid are they?

Is it possible to exploit a process like non linear self-focussing to change beam parameters during the pulse time? By tuning the intensity over time envelope, what might be done by apropriate chirping, the effective "focal length" of the non linear material can be preprogrammed. (But not be controlled in real time)

Dear Luis,

Sorry to reply late. What i need is predefined sequence of wavefront.  The pulse is narrowband, several nanoseconds width,  1053nm wavelength. And bandwidth can be up to several nm if phase modulation is necessary. The focal spot is shaped to be hundreds um large from diffraction limit using phase plate.

Dear Alexander Draebenstedt,

According to my understanding，what you said is self phase modulation(SPM), but the waveform of the pulse is fixed, it can't be changed. But similarly, we can add another beam to achieve cross phase modulation(XPM). It's a possible way. Thank you to remind me.

The easiest way seems to me building something like a mach zehnder interferometer, add some delay to one arm, and induce a static width or convergence change between arms. Upon recombination they'll focus differently, but the overall pulse length will change, which I don't know if it is acceptable.

Alternatively, if you have plenty of power, you could research on Kerr lensing. It's similar to filamentation, but before you lose control :)