yes, of course in general it does, as long as the group velocity is considered. Usually, this gives a very small mismatch contribution, but with very short pulses, also because of that, very thin crystals must be used.

Given the "monochromatic" in the question I would tend to ignore the dispersion within the excitation bandwidth. In that case the answer is "no". The phase shift between the fundamental and second harmonic is imprinted in the third harmonic but has no effect on the efficiency (total amount THG).

There are however some potential pitfalls here. If the SHG is done using two different polarizations of the fundamental, the output polarization state of the fundamnetal is not defined (depends on the exact phasehshift between the two axes which is not well defined and can easily shift with temperature. basically the crystal will act as a very thick Lambda plate of badly defined thickness). If you do SHG (and THG) in something like PPLN, all polarizations are aligned and you will be fine. Also birefringent comversion using eeo interaction (where the SHG has a different polerization from the fundamantal but both fundamnetal photons are the same) is OK.

This however means that you go into the THG with two different polarizations which may not be the most efficient configuration (but you need a very complex lambda plate to change one polarization without changing the other).

One thing to consider carefully is that the amount of conversion in the first stage is quite important for the efficiency in teh second stage, you want to end up with the right balance between first and second harmonic after the first stage which (unfortunately) is power dependent. People have looked at doing SHG and THG in PPLN simultaneously (with engineered relative conversion strengths so that you get maximum conversion over the length of the crystal.

Sure! I agree for "monochromatic", i.e. "long pulses".

Thank you, Herman and Carlo, for your interest and answers. Let me mention, that nonexpensive double wavelength waveplates with very good quality for custom wavelengths are avaliable at Day Optics (www.dayoptics.com).

In the case of few cycle pulses with high intensity, when usually used

the thin crystal, it should be taken into account that during the propagation of the pulses the phase relation between phases of interacted pulses mainly determined by nonlinearity of interaction and the phase changes caused by dispersion can be neglected. The efficiency of harmonic generation depend from phase relation of interacted pulses at the input of crystal.

To my mind, correct answer is not obvious. Could you refer to any publications ?

I tried to check it experimentally, setting different distances between SGH and THG crystals, using Q-switched YAG: Nd laser and LBO crystals (the both for oo-e) . Dispersion in air will change phase shift. But efficiency was almost constant (within 10% error).

David, it was your paper with THG theory ?

At first please approve if you work with few cycle duration laser pulses (what is duration of pulse ?)

Notes in my previous answer was based on the results of numerical simulations,

related to the numerical calculation of Maxwell's equations by FDTD.

In simulations considered propagation of few-cycle duration pulses through the nonlinear medium with second order nonlinear susceptibility.

When you speak about the phase shift in air you should be consider the influence of beams spatially distributions on the harmonic generation efficiency also.

Besides if you consider the few-cycle duration laser pulses you should be take into account the dispersion of reflection/transmission at the input of the crystal also.

I agree that few cycle pulses have a bandwidth that is so large that nonlinear effects create phase changes within the bandwidth of the pulse (SPM, FWM).

Q-switched Nd:YAG pulses however are generally a few nsec to tens of nsec long. They thus contain something like one million cycles and the dispersion within the bandwidth is very low, even for crystal that are tens of mm long. I think the experiment (changing the crystal position) shows what you would expect for this case: that there is little change. With these kind of pulses you should be more worried about beam walk-off; the reduction of overlap between fundamental and SHG due to the difference between the k-vectors and Poynting vector in the birefringent crystal. You can compensate the walk-off (and improve the beam profile and conversion) by using a pair of crystals with their axes reversed.

I too agree that the phases of the fundamental and the second harmonic should not be an issue. In fact, for the SFG/THG crystal, it is equivalent to have two beams from two different lasers. So, I would expect that the efficiency should not change with distance in this case. However, if one uses two or more nonlinear crystals to enhance the SHG efficiency, the situation will be different. Because, in this case the condition of non-zero input at second harmonic frequency is no longer valid at the input to the second crystal. So, the phase relation between the fundamental and the second harmonic after the first crystal will effect the efficiency and hence the distance between the subsequent stages might effect the efficiency. This is what we found while trying to improve the SHG efficiency in CW using multiple crystals. http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-19-12-11152

You are absolutely right concerned SHG. This item had been considered in past years for intracavity SHG to set correct distance between SHG crystal and mirror, that returns both beams back after forward pass.

I suppose, in cascade THG, when THG efficiency is low, significant periodical dependance of phase retardation must exist, because A*3 ~A*1 x A*2. Here we can not consider A1 and A2 as independent beams from 2 different sources. When efficiency increases, and we must consider pump depletion, TH periodical modulation reduces and can be masked by another factors: instability of laser energy, beam size variation with distance, etc., that`s why at practically important conditions we cannot detect it clear.

To maximize THG of nanosecond singlemode beams with gaussian radial profile and beam diameter >1 mm, according my experience, the most important is correct angular positioning of SHG crystal, because we can not obtain optimum relation between intensities of SH and fundamental mode over all beam diameter at the same time. On practice we measure integrated (over beam diameter) value, and this value never exceed 40-45%. If you set crystal angle to get maximum SHG, then THG efficiency reduces ~1.5 times.

Agreed! This is quite interesting to know. When I said that the fundamental and the SHG can be considered to be coming from two different sources, I meant from the THG crystal point of view. When I did this experiment in CW, it was always limited by the number of photons at the SH wavelength, because of the efficiency (36%) in SHG stage. I observe a direct correlation.

In the ns regime, do you think that the reduction in THG efficiency by 1.5 times when the SHG efficiency is maximized mean that the phase relationship between fundamental and SHG is channging with SHG efficiency? Could it be due to thermal effects?

Let me explain more detail: we deal with gaussian beams. As energy density depends on radius, so SHG efficiency also depends on radius. It leads to the distortion of fundamental mode profile after SHG. At low input energy , when SHG efficiency does not exceed ~10%, distortion is minimal, but at high input level the rest energy density in the center of fundamental beam will be lower than at some distance from the center.

Relation between local energy density of fundamental beam and SH, required for THG, is 1: 2. For Gaussian beams (or any other beams with non flat profile) this relation can be realized only at some radius (for simplicity I suppose absolutely axially symmetric profile). And total (integrated over beam spot) efficiency will result from ovelapping of intensity profiles of fundamental mode and SH.

To obtain the best ovelapping, at low SHG efficiency , of course you must set crystal angle for best matching. But at high input you need to detune the crystal to restore central part of fundamental beam. In my case maximum THG efficiency was observed, when crystal was detuned by ~0.25 of angular acceptance angle from best matching position.

In fact, to obtain maximum possible conversion efficiency we must use beams with flat (supergaussian) profiles as well as satisfy other well known conditions (small angular divergence, narrow spectral width, polarization matching, etc.). Efficiences ~80-90% were obtained in 1980-90 just with supergaussian beams.

Mention, that physical processes in main features are enough clear, but nobody developed a completed theory yet.

The discussed dependance of THG efficiency on phase retardation between 1 and 2 harmonics at TH crystal input is another question.

Hi

I am trying to have a setup for third harmonic generation using bbo

i can have up to 200 mw conversion from bbo at 400 nm, then putting

an other crystal BBO (equal) i don t see any third harmonic generation.

in what I am wrong ?

I have to put also the fundamental frequency on the same crystal ?

Thanks

To obtain 3 harmonic you can mix 1 harmonic and 2 harmonic of your laser. The most common way is to obtain 2 harmonic using crystal A, and then mix it with the rest of 1 harmonic in crystal B. Of course, crystal A and crystal B angles and orientation will not be the same. You must consider the type of harmonic generation (OO-E or OE-E) and, may be, use polarisation-matching components for 3 harmonic generation.

respected Aleksandr sir, what is the reason regarding that if the second harmonic generation efficiency is increased than why third harmonic generation susceptibility is decreases?

Dear Jaydeep Joshi, I suppose you mean "third harmonic efficiency" instead "succeptibility" ?

So, I only repeat, that we obtain 3 harmonic by mixing 2 harmonic and the rest amount of 1 harmonic. Optimum SHG efficiency here is 66%. In this case you get the equal number of photons of 1 and 2 harmonic at the input of THG crystal, and theoretically you can get 100% THG efficiency.  If you have larger SHG efficiency, for example, 90%, then the rest amount of first harmonic is 10% (of energy). In this case maximum possible THG efficiency should be 30%.

Mention, that here I only answer for your question and do not consider problems related to gaussian beams.

Dear colleagues,

while thinking about optimizing the THG efficiency in one of our pulsed dye laser systems, I stumbled upon this thread which I read with great interest. My background is in fact engineering and not physics, though I've worked with lasers for about 10 years now so I think I can say that I have a reasonably good understanding of the underlying processes (while sometimes lacking theoretical physical background). I realize that this thread is already quite old, but maybe there are still some people actively read new posts here.

At the moment, we have a cascaded THG setup (SHG with type-I BBO and subsequent sum frequency generation in another, differently cut, BBO crystal). Before the SHG stage, a half-wave plate is placed in the fundamental beam path to tune the polarization for maximum THG output. This is approximately 30° away from vertical polarization. SHG output is horizontally polarized, so the BBO for SFG is rotated 90°.

Now my question: I guess, the optimum case for SFG is - just as it is for THG - that the polarization of the fundamental and the SHG beam is parallel? In fact, the polarisation of the leftover fundamental beam after the SHG is almost perfectly circular (which I guess it would be at perfectly vertical polarization). How strong does this affect the THG efficiency? From what I suspect, only the component parallel to the SHG polarization can be used for THG conversion, i.e. a large portion of the fundamental beam passes the second crystal without interaction.

If I were to convert the polarization back to linear and match it with the SHG polarization, would that have a positive impact on THG efficiency? As the beams spatially overlap, I would need to separate them, add a quarter-wave plate (for conversion to linear polarization) and a half-wave plate (for rotation of polarization) in the fundamental beam, and combine them again before the SHG.

I guess I'll have to keep the distances short (or delay the other beam) to keep temporal mismatch minimal. Pulse length is 10 ns.

I appreciate any comments and thoughts on that idea.

Thanks

Max

Ruslan Subkhangulov Thank you for your answer.

And the information to everyone interested: I just tried it out and it works brilliantly. Instead of a complicated beam path, I simply ordered specialized retardation plates. So the setup is as follows:

Fundamental with vertical polarization

SHG BBO

Retardation plate l/4 for fundamental and l/1 for SHG

Retardation plate l/2 for fundamental and l/1 for SHG

THG BBO

With the l/4 plate, linear polarization is recovered, with the l/2 plate, polarization is aligned with the horizontal polarization of the SHG beam. Both retardation plates do not affect polarization of the SHG beam.

They have a quite narrow acceptance range for them to work in spec, maybe a few nm.

But this improved output (with a couple of minutes for adjusting) from 6-7 mJ to over 10 mJ at 230 nm.