Before all, you may need to understand the self-focusing effect in media. When a Gaussian shaped beam transmits a medium (in this case, Ti:sapphire crystal), the refractive index of the medium changes based on the equation n(I) = n0 + n2I where I is the intensity of the incoming beam, n0 is the refractive index, and n2 is the "nonlinear refractive index". It should be noted that this nonlinear refractive index change occurs with any materials (any crystals, any glasses). Since a Gaussian beam has the intensity distribution (the highest at center), the induced refractive index change also has a Gaussian distribution. This spatially distributed refractive index acts as a lens. Because n2 is always positive, the lens is positive, so the incoming beam is focused by this virtual lens which is induced by the beam itself. This is reason why this phenomenon is called "self" focusing.

Please note that the nonlinear refractive index change n2 is so small, e.g. ~3 x 10^(-16) cm2/W for sapphire. So a self-focusing happens only when the incoming beam has very high peak power.

In short, Kerr-lens modelocking is based on the difference in loss (or gain) for continuous-wave (CW) and pulsed operation. If a Ti:sapphire laser is arranged so that the cavity exhibits lower loss (or higher gain) when a self-focusing happens, the laser tends to operate in mode-locked operation by itself.

There are, in general, two types of Kerr-lens modelocking.

One is called "hard-aperture" modelocking. Here, "hard aperture" represent a slit in a laser cavity which introduces a loss. The point is that the amount of loss due to the slit depends on the beam size. In this case, the laser cavity is arranged so that the beam size becomes smaller when the self-focusing happens.

The other is called "soft-aperture" modelocking. In this case, the variation of gain rather than loss contributes to mode-locking. The laser gain depends on the overlap between cavity mode and pump beam (so called mode-matching efficiency). If the laser cavity is arranged so that the mode-matching efficiency increases with the self-focusing effect, the laser tends to operate with higher gain.

Dear Hiroki,

Thank you for your valuable information. You answer this question completely and this shows that you have a lot of experience and knowledge about this topic. For this reason I ask another question.

You said " Kerr-lens modelocking is based on the difference in loss (or gain) for continuous-wave (CW) and pulsed operation. " and the question that arise here is:

Are there both CW and Pulsed regimes in Ti: sapphire lasers in the same time and slit separate them from each other?

Or

There is only one regime (pulsed regime) and this shows a Gaussian shape pulse that face more loss in head and tail ends and less loss in center and acts as a saturable absorption e.g slit separate the low intensity part of Gaussian pulse from High intensity part.

Could you please describe this specific point.

Thanks again

Dear Aydin,

Your latter answer is correct. The combination of self-focusing and slit can be considered as a virtual saturable absorber.

The CW and pulsed (modelocked) modes do not exist at the same time.

Lasers operates in one of two regime.

It would be worthwhile to know about the phase relation of longitudinal modes. "Mode" of the term "modelocking" means longitudinal mode. When a laser is modelocked, the phase of longitudinal modes are locked (or, let's say, correlated). Actually, in a free running CW Ti:sapphire laser, numerous longitudinal modes oscillate. The CW output is resulting from the superposition of the longitudinal modes with no phase correlation.

Thus, the difference between CW and pulsed modes is only the relative phase among longitudinal modes. This may explain why a laser operates one of two regimes at the same time.

Please feel free for further questions.

Dear Rotem,

Thank you for sharing this article.

Dear Hiroki,

Thank you for your response.

I was in doubt, and you've fixed it.

Why this figure only shows d1 and d2, distances between Ti:sapphire and mirrors 2 and 3 and do not show distances elsewhere?

Are there any point here?

It might help to back up and explain the basic premise. When all of the modes, which have different frequencies, happen to be in phase at one moment you get a very high intensity. Because the modes quickly dephase you also get a very short pulse.

If you can make a laser resonator that favors very high intensity (more round trip gain, less round trip loss) then the very high intensity mode locked condition will be favored and mode locking can be sustained. (the cavity also must have low dispersion so the cavity itself doesn't dephase the modes.)

To make a cavity that will favor very high intensity you look for any property that is different at high intensity, so you look for strong non-linear effects. One non-linear effect is Kerr lensing. A high intensity beam will experience strong lensing in many materials due to the intensity profile.

Now the question is, how can you use the extra lensing to make a cavity favor high intensity? Well, lensing affects the mode size. The strongest mode size dependent effect is gain overlap. If you focus the pump beam and make the gain stripe in the gain medium very small, then the size of the resonating beam through the gain medium will have a very strong effect on how much gain you get. If the cavity is arranged so that you only get a very small mode in the gain medium when Kerr lensing is included, then very intense modes will experience higher gain and mode locking will be favored. Another mode size dependent effect is loss at an aperture. If you have a hard aperture and the mode size is smaller at the aperture when Kerr lensing is included, then very intense modes will experience less loss at the hard aperture and mode locking will be favored. An adjustable slit is used so that the aperture size can be adjusted to give the maximum difference between the loss for low intensity modes and high intensity modes.

Dear Rotem,

Thank you for your response

Dear Mike,

Thank you for your response.

I accept what you and others mentioned here and actually we all are in the same boat. However, there is a small challenging question that bother me and I don't have any reasonable answer to it, and it is:

By removing small intensities, actually we remove the frequencies. Am I right?

As we know from the Heisenberg uncertainty principle, by reducing frequency we expect to have greater time (but only with this slit we can reach < 10 femtosecond pulses). How we can describe this?

I think I found the answer of my last question. However, I wait to others to share their knowledge with us.

Thanks a lot.

No, I don't think favoring high intensity removes frequencies. Your own Heisenberg argument shows why. More frequencies gives a narrower and more intense pulse. Higher intensity is achieved by a broader spectrum, so why would you think favoring high intensity narrows the spectrum? It's quite the opposite.

Sorry for being a pest. I want to understand this phenomena deeply because in a meeting with my friends there are different descriptions related to it.

Do we remove some frequencies by blocking low intensities (wings of pulses) using adjustable slit (or in general with saturable absorption)?

The answer is yes, we remove some frequencies of pulse at the tailing and leading part of it. However, this (KLM accompanying with adjustable slit) is the only way that we can reach under 10 fs pulses.

I think there is another description that we should bring here. maybe we should change our point of view.

Thank you all to participating in this topic.

What the slit is selecting is more Kerr lensing vs less Kerr lensing. More Kerr lensing produces a smaller transverse mode size at the slit. This means that a higher intensity pulse will suffer less loss at the slit than a lower intensity pulse. This selects for the mode locked condition.

Saying the same thing backwards: the highest possible intensity with the highest possible Kerr lensing occurs when the broadest spectrum of frequencies (limited by the gain spectrum of the medium) are all in lock step and periodically come into phase producing a short super intense pulse. This maximum intensity pulse experiences the maximum possible Kerr lensing which produces a focus at the slit so it passes through with little loss. Any less intense pulse will not focus at the slit and will suffer significant loss.

Dear Mike

I appreciate your answer. You said what we want.

Thanks again.

You can place a slit in the cavity to exploit the self-focusing effect and use it as the mode-locking mechanism, but there is actually a more subtle effect. The mode volume in the Ti:sapphire rod will change when the intensity is high enough to induce self-focusing, and therefore there will be a change of the spatial overlap between the pumped volume, and the laser mode volume, changing the gain and therefore producing the mode-locking effect. The slit in modern Ti:sapphire lasers is used to limit bandwidth and select central wavelength.

I believe Mr. Rangel-Rojo has it right. I was mistaken to say the slit is used for mode-locking. As I said, the typical method for generating self mode-locking is using the Kerr lensing to increase the mode overlap in the gain medium ("soft aperture KLM"). While it is also possible to use a slit to cause mode-locking as I described ("hard aperture KLM"), apparently that is not how the slit is used in commercial Ti:sapphire lasers. I apologize. I am not very up to date on Ti:Sapphire lasers. The only one with which I am intimately familiar is one I built myself in the early nineties shortly after ultra-short pulses were first shown and before commercial units were available.

I guess the question is whether you want to understand how a fs Ti:Sapp works or whether you want to know how to get a Ti:Sapp to produce fs pulses. For the cw case, the distance between the two curved mirrors is adjusted so you get the maximum output - and this happens when the cavity is 'most stable'. To get fs pulses you have to change this distance as the Kerr effect produces some additional focussing. This obviously means that the cw power goes down. And to start the mode locking process, you need high intensities - this is usually achieved by tapping the table so the mirrors vibrate slightly, and you introduce a noise spike. Commercial system obviously use more sophisticated ways of starting the mode locking - but still: even commercial systems need some means to start the mode locking process. Once you have pulses, the laser might stay mode locked for several hours.

Dear Raul

Thank you for your response. could you tell us more about this sentences" changing the gain and therefore producing the mode-locking effect ".

How self focusing impact on gain or inversely how gain impact on self focusing ?

Bests,

Dear Mike

Your information was worthy to us.

Thank you.

Dear Rainer,

Thank you for your practical and empirical information that would be valuable to us.

I never mentioned the word 'gain'. I was talking about the resonator configuration: when the cavity is 'most stable' you will have the lowest losses (e.g. due to diffraction). The stability criterion is different with and without Kerr lensing. So when you optimise the cavity for cw operation (most stable = highest output power) then the distance between the two curved mirrors is not optimal for pulsed operation => to favour pulsed operation you will have to change this distance. I agree with a previous response that the overlap between your pump beam and the cavity beam changes when you have mode locking but that would be only a second order effect.

Dear Rainer,

I replied the word "gain" for Dr. Raul Rangel-Rojo's response in this sentence " The mode volume in the Ti:sapphire rod will change when the intensity is high enough to induce self-focusing, and therefore there will be a change of the spatial overlap between the pumped volume, and the laser mode volume, changing the gain and therefore producing the mode-locking effect."

However, I couldn't understand how the overlap between pump and signal persuade modes to lock.

Bests,

Exculse me, I reviewed the answers and found out that Dr. Mike answered my last question in his second response.

As I see it is that whn yoy change the overlap, the gain diminishes, which you can see it as you introduce losses, and these losses are the ones responsible for the saturable absorber effect that produces mode-locking. Because this is produced by the Kerr effect has an instantaneous response time, it allows you to produce the shortest possible pulses. There are some talks available in the internet by Prof. Ursula Keller, that was the first to explain this 'magical' mode-locking

Dear Raul,

Thank you for your response. I got the point. I will find that talks and will study them.

Best regards,