# I Atomic units of frequency and angular frequency

1. Aug 5, 2016

### Ecthe

To calculate the Keldysh parameter, I need to use the optical frequency of a laser in atomic unit.

Since for the time: 1 a.u. = 2.42×10-17 s, I would assume that for the frequency:
1 a.u. = 4.13×1016 s-1 which is juste the inverse of the time one. However, I found several sources such as:
http://www.diss.fu-berlin.de/diss/s...ISS_derivate_000000001950/08_app_C.pdf?hosts=
and
http://web.ift.uib.no/AMOS/PHYS261/2008/LATEX-work/AtomicUnits/Lene_Raymond_Abdul.pdf
where for the frequency: 1 a.u. = 6.58×1015 s-1 which correspond to: 4.13×1016 s-1 / (2π).

So I would assume that by frequency, they mean angular frequency and then it would make sense to me. But maybe I overlooked something and that is why I am posting here.

Thanks in advance for any help.

2. Aug 5, 2016

### Staff: Mentor

That second reference explicitly mentions that this is the number for angular frequency.

What is important is what goes into the formula you are using. What are you using to calculate the Keldysh parameter?

3. Aug 5, 2016

### Ecthe

Thanks for your answer. Indeed, on the second reference, I was looking at the table 3 and missed the statement above.

To calculate the Keldysh parameter, I use the folowing formula:
γ=ω(2Ip)1/2/E
Where:
ω: the laser angular carier frequency
Ip: atom ionization potential
E: laser electric field
All of those in atomic units.

4. Aug 8, 2016

### IgnasB

Hello,

I recently calculated Keldysh parameter in order to model ionization rates of gas interacting with intense laser fields.

I used slightly different formula:
γ = ω (2Ip)1/3/F.​
You can calculate ω, for example, like this - 2π(c/λ)/4.13e16, where c is the speed of light and λ is the wavelength of light.

Ip is usually in eV, and to covert it to atomic units you need to divide Ip/(a. u. of energy), which is 27.21 eV; i.e. [2*H(hydrogen) ground state binding energy], I_p_hydrogen = 13.6 eV, hence the 2 before Ip.

F also has to be in atomic units, F symbolizes electric field at the point of interest and atomic unit of field is 5.1422e11 V/m.

So the whole formula should look like this:

γ = (2π(c[m/s]/λ[m])/4.13e16[s-1]) * (2Ip[eV]/27.21[eV])1/3/(F [V/m]/5.1422e11 [V/m]).

Hope that helps.

5. Aug 8, 2016

### Ecthe

Hello, thanks for your reply. Yes I actually made a mistake in the formula and it is (2Ip)1/3 and not (2Ip)1/2.

I am calculating it for the exact same purpose. Thus, I would very much like to exchange with you on that matter. The reason for that is that I had a look at several papers to calculate the ionization rate and the formula given are not always consistent with each other.

6. Aug 8, 2016

### IgnasB

Yes, I know what you mean, like every dissertation or book, or paper I took, had slightly different formulations of the same formulas.

Which theory did you use (or plan to use) - ADK, PPT, TDSE or something else?

For my purposes I settled with ADK (when Keldysh parameter is < 0.5), and I found a very clean paper where formulas were clearly written out (2.2 section):

2012 Yan-Zhuo; -- Multiphoton and tunneling ionization of atoms in an intense laser field. (pm me if you can't find/download this paper)

or you can check out a book named:

Fundamentals of Attosecond Optics by Z. Chang where you can find calculated constants for ADK formula for different noble gases (and of course the formulas themselves).

7. Aug 8, 2016

### Ecthe

Right now I am using the ADK one. For the papers, I had a look at several ones :

-The original one from Ammosov et al. (1986).
-Tong & Lin (J. Phys. B, 2005). In that one, they developp an empirical formula which allow they modified ADK rate to fit the TDSE even for higher electrical field strengh. However, they also provide the non modified ADK formula.
-Yan-Zhuo (2012) in wich their ADK formula is the exact same as the one in Fundamentals of Attosecond Optics
-Fundamentals of Attosecond Optics

Mainly, I tried to write all of them with the same formulations and end up with slight differences even after checking my calculations. But it is always hightly possible that I made a mistake. So to be sure my calculations and code are correct, I am currently plotting (like really as I am writting here...) my results on the top of their plots.

8. Aug 8, 2016

### IgnasB

I tried to do the same with the plot for Neon from Yan-Zhuo paper. Got very similar results, like very very similar, but I checked only by eye (zoomed in as close as I could). But they could have used like 3e8 for the speed of light, not 299... , or some slightly different ionization potential and we probably will never know.. But, please, tell me how your calculations went after you fit those graphs, I am very interested.

I plan myself to model another formula like PPT and then check if the ionization rate values are similar to ADK ones. But probably only towards the end of the week, since I have some work to do at the lab.